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Unformatted text preview: 5E-13(pp 828-837) 1/18/06 11:09 AM Page 828 CHAPTER 13 Wind velocity is a vector because it has both magnitude and direction. Pictured are velocity vectors indicating the wind pattern over San Francisco Bay at 12:00 P.M. on June 11, 2002. Vectors and the Geometry of Space 5E-13(pp 828-837) 1/18/06 11:09 AM Page 829 In this chapter we introduce vectors and coordinate systems for three-dimensional space. This will be the setting for our study of the calculus of functions of two variables in Chapter 15 because the graph of such a function is a surface in space. In this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space. |||| 13.1 Three-Dimensional Coordinate Systems z O y x FIGURE 1 Coordinate axes z y To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair ͑a, b͒ of real numbers, where a is the x-coordinate and b is the y-coordinate. For this reason, a plane is called two-dimensional. To locate a point in space, three numbers are required. We represent any point in space by an ordered triple ͑a, b, c͒ of real numbers. In order to represent points in space, we first choose a fixed point O (the origin) and three directed lines through O that are perpendicular to each other, called the coordinate axes and labeled the x-axis, y-axis, and z-axis. Usually we think of the x- and y-axes as being horizontal and the z-axis as being vertical, and we draw the orientation of the axes as in Figure 1. The direction of the z-axis is determined by the right-hand rule as illustrated in Figure 2: If you curl the fingers of your right hand around the z-axis in the direction of a 90Њ counterclockwise rotation from the positive x-axis to the positive y-axis, then your thumb points in the positive direction of the z-axis. The three coordinate axes determine the three coordinate planes illustrated in Figure 3(a). The xy-plane is the plane that contains the x- and y-axes; the yz-plane contains the y- and z-axes; the xz-plane contains the x- and z-axes. These three coordinate planes divide space into eight parts, called octants. The first octant, in the foreground, is determined by the positive axes. z x z FIGURE 2 Right-hand rule y z-plan ne la xz-p x FIGURE 3 e l al ft w O xy-plane (a) Coordinate planes le y x right w all O floor y (b) Because many people have some difficulty visualizing diagrams of three-dimensional figures, you may find it helpful to do the following [see Figure 3(b)]. Look at any bottom corner of a room and call the corner the origin. The wall on your left is in the xz-plane, the wall on your right is in the yz-plane, and the floor is in the xy-plane. The x-axis runs along the intersection of the floor and the left wall. The y-axis runs along the intersection of the floor and the right wall. The z-axis runs up from the floor toward the ceiling along the intersection of the two walls. You are situated in the first octant, and you can now imagine seven other rooms situated in the other seven octants (three on the same floor and four on the floor below), all connected by the common corner point O. 829 5E-13(pp 828-837) 830 ❙❙❙❙ 1/18/06 11:09 AM Page 830 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z Now if P is any point in space, let a be the (directed) distance from the yz-plane to P, let b be the distance from the xz-plane to P, and let c be the distance from the xy-plane to P. We represent the point P by the ordered triple ͑a, b, c͒ of real numbers and we call a, b, and c the coordinates of P; a is the x-coordinate, b is the y-coordinate, and c is the z-coordinate. Thus, to locate the point ͑a, b, c͒ we can start at the origin O and move a units along the x-axis, then b units parallel to the y-axis, and then c units parallel to the z-axis as in Figure 4. The point P͑a, b, c͒ determines a rectangular box as in Figure 5. If we drop a perpendicular from P to the xy-plane, we get a point Q with coordinates ͑a, b, 0͒ called the projection of P on the xy-plane. Similarly, R͑0, b, c͒ and S͑a, 0, c͒ are the projections of P on the yz-plane and xz-plane, respectively. As numerical illustrations, the points ͑Ϫ4, 3, Ϫ5͒ and ͑3, Ϫ2, Ϫ6͒ are plotted in Figure 6. P(a, b, c) c O a y x b FIGURE 4 z z z 3 (0, 0, c) R(0, b, c) S(a, 0, c) 0 P(a, b, c) _5 x (_4, 3, _5) (0, b, 0) (a, 0, 0) 3 _2 y y x 0 0 _4 _6 y x (3, _2, _6) Q(a, b, 0) FIGURE 5 FIGURE 6 Խ The Cartesian product ‫ ޒ‬ϫ ‫ ޒ‬ϫ ‫͕͑ ෇ ޒ‬x, y, z͒ x, y, z ʦ ‫ ͖ޒ‬is the set of all ordered triples of real numbers and is denoted by ‫ .3 ޒ‬We have given a one-to-one correspondence between points P in space and ordered triples ͑a, b, c͒ in ‫ .3 ޒ‬It is called a threedimensional rectangular coordinate system. Notice that, in terms of coordinates, the first octant can be described as the set of points whose coordinates are all positive. In two-dimensional analytic geometry, the graph of an equation involving x and y is a curve in ‫ .2 ޒ‬In three-dimensional analytic geometry, an equation in x, y, and z represents a surface in ‫.3 ޒ‬ EXAMPLE 1 What surfaces in ‫ 3 ޒ‬are represented by the following equations? (a) z ෇ 3 (b) y ෇ 5 SOLUTION Խ (a) The equation z ෇ 3 represents the set ͕͑x, y, z͒ z ෇ 3͖, which is the set of all points in ‫ 3 ޒ‬whose z-coordinate is 3. This is the horizontal plane that is parallel to the xy-plane and three units above it as in Figure 7(a). z z y 5 3 0 x FIGURE 7 0 (a) z=3, a plane in R# y x 5 (b) y=5, a plane in R# 0 y (c) y=5, a line in R@ x 5E-13(pp 828-837) 1/18/06 11:09 AM Page 831 SECTION 13.1 THREE-DIMENSIONAL COORDINATE SYSTEMS ❙❙❙❙ 831 (b) The equation y ෇ 5 represents the set of all points in ‫ 3 ޒ‬whose y-coordinate is 5. This is the vertical plane that is parallel to the xz-plane and five units to the right of it as in Figure 7(b). NOTE When an equation is given, we must understand from the context whether it represents a curve in ‫ 2 ޒ‬or a surface in ‫ .3 ޒ‬In Example 1, y ෇ 5 represents a plane in ‫ ,3 ޒ‬but of course y ෇ 5 can also represent a line in ‫ 2 ޒ‬if we are dealing with two-dimensional analytic geometry. See Figure 7(b) and (c). In general, if k is a constant, then x ෇ k represents a plane parallel to the yz-plane, y ෇ k is a plane parallel to the xz-plane, and z ෇ k is a plane parallel to the xy-plane. In Figure 5, the faces of the rectangular box are formed by the three coordinate planes x ෇ 0 (the yz-plane), y ෇ 0 (the xz-plane), and z ෇ 0 (the xy-plane), and the planes x ෇ a, y ෇ b, and z ෇ c. z y 0 ■ EXAMPLE 2 Describe and sketch the surface in ‫ 3 ޒ‬represented by the equation y ෇ x. SOLUTION The equation represents the set of all points in ‫ 3 ޒ‬whose x- and y-coordinates Խ x are equal, that is, ͕͑x, x, z͒ x ʦ ‫ ,ޒ‬z ʦ ‫ .͖ޒ‬This is a vertical plane that intersects the xy-plane in the line y ෇ x, z ෇ 0. The portion of this plane that lies in the first octant is sketched in Figure 8. FIGURE 8 The plane y=x The familiar formula for the distance between two points in a plane is easily extended to the following three-dimensional formula. Խ Խ Distance Formula in Three Dimensions The distance P1 P2 between the points P1͑x 1, y1, z1 ͒ and P2͑x 2 , y2 , z2 ͒ is Խ P P Խ ෇ s͑x 1 z P¡(⁄, ›, z¡) Ϫ x 1 ͒2 ϩ ͑y2 Ϫ y1 ͒2 ϩ ͑z2 Ϫ z1 ͒2 2 To see why this formula is true, we construct a rectangular box as in Figure 9, where P1 and P2 are opposite vertices and the faces of the box are parallel to the coordinate planes. If A͑x 2 , y1, z1͒ and B͑x 2 , y2 , z1͒ are the vertices of the box indicated in the figure, then P™(¤, fi, z™) ԽP AԽ ෇ Խx 1 2 Ϫ x1 Խ Խ AB Խ ෇ Խ y Ϫ y1 2 Խ Խ BP Խ ෇ Խ z 2 2 Ϫ z1 Խ Because triangles P1 BP2 and P1 AB are both right-angled, two applications of the Pythagorean Theorem give 0 x 2 B(¤, fi, z¡) ԽP P Խ ԽP BԽ 1 A(¤, ›, z¡) y and 2 2 1 FIGURE 9 2 Խ Խ ෇ ԽP AԽ ෇ P1 B 2 2 1 Խ Խ ϩ Խ AB Խ ϩ BP2 2 2 Combining these equations, we get ԽP P Խ 1 2 2 Խ Խ ϩ Խ AB Խ ϩ Խ BP Խ ෇ Խx Ϫ x Խ ϩ Խy Ϫ y Խ ϩ Խz ෇ P1 A 2 2 2 1 2 2 2 1 2 2 2 Ϫ z1 Խ 2 ෇ ͑x 2 Ϫ x 1 ͒2 ϩ ͑y2 Ϫ y1 ͒2 ϩ ͑z2 Ϫ z1 ͒2 Therefore Խ P P Խ ෇ s͑x 1 2 2 Ϫ x 1 ͒2 ϩ ͑y2 Ϫ y1 ͒2 ϩ ͑z2 Ϫ z1 ͒2 5E-13(pp 828-837) 832 ❙❙❙❙ 1/18/06 11:10 AM Page 832 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 3 The distance from the point P͑2, Ϫ1, 7͒ to the point Q͑1, Ϫ3, 5͒ is Խ PQ Խ ෇ s͑1 Ϫ 2͒ 2 ϩ ͑Ϫ3 ϩ 1͒2 ϩ ͑5 Ϫ 7͒2 ෇ s1 ϩ 4 ϩ 4 ෇ 3 EXAMPLE 4 Find an equation of a sphere with radius r and center C͑h, k, l͒. z SOLUTION By definition, a sphere is the set of all points P͑x, y, z͒ whose distance from C is r. (See Figure 10.) Thus, P is on the sphere if and only if PC ෇ r. Squaring both sides, we have PC 2 ෇ r 2 or P(x, y, z) Խ Խ Խ Խ r ͑x Ϫ h͒2 ϩ ͑y Ϫ k͒2 ϩ ͑z Ϫ l͒2 ෇ r 2 C(h, k, l) The result of Example 4 is worth remembering. 0 Equation of a Sphere An equation of a sphere with center C͑h, k, l͒ and radius r is x ͑x Ϫ h͒2 ϩ ͑y Ϫ k͒2 ϩ ͑z Ϫ l͒2 ෇ r 2 y FIGURE 10 In particular, if the center is the origin O, then an equation of the sphere is x 2 ϩ y 2 ϩ z2 ෇ r 2 EXAMPLE 5 Show that x 2 ϩ y 2 ϩ z 2 ϩ 4x Ϫ 6y ϩ 2z ϩ 6 ෇ 0 is the equation of a sphere, and find its center and radius. SOLUTION We can rewrite the given equation in the form of an equation of a sphere if we complete squares: ͑x 2 ϩ 4x ϩ 4͒ ϩ ͑y 2 Ϫ 6y ϩ 9͒ ϩ ͑z 2 ϩ 2z ϩ 1͒ ෇ Ϫ6 ϩ 4 ϩ 9 ϩ 1 ͑x ϩ 2͒2 ϩ ͑y Ϫ 3͒2 ϩ ͑z ϩ 1͒2 ෇ 8 Comparing this equation with the standard form, we see that it is the equation of a sphere with center ͑Ϫ2, 3, Ϫ1͒ and radius s8 ෇ 2s2. EXAMPLE 6 What region in ‫ 3 ޒ‬is represented by the following inequalities? 1 ഛ x 2 ϩ y 2 ϩ z2 ഛ 4 zഛ0 SOLUTION The inequalities z 1 ഛ x 2 ϩ y 2 ϩ z2 ഛ 4 can be rewritten as 1 ഛ sx 2 ϩ y 2 ϩ z 2 ഛ 2 0 1 2 x FIGURE 11 y so they represent the points ͑x, y, z͒ whose distance from the origin is at least 1 and at most 2. But we are also given that z ഛ 0, so the points lie on or below the xy-plane. Thus, the given inequalities represent the region that lies between (or on) the spheres x 2 ϩ y 2 ϩ z 2 ෇ 1 and x 2 ϩ y 2 ϩ z 2 ෇ 4 and beneath (or on) the xy-plane. It is sketched in Figure 11. 5E-13(pp 828-837) 1/18/06 11:10 AM Page 833 SECTION 13.1 THREE-DIMENSIONAL COORDINATE SYSTEMS |||| 13.1 ❙❙❙❙ 833 Exercises 1. Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position? 2. Sketch the points ͑0, 5, 2͒, ͑4, 0, Ϫ1͒, ͑2, 4, 6͒, and ͑1, Ϫ1, 2͒ on a single set of coordinate axes. 3. Which of the points P͑6, 2, 3͒, Q͑Ϫ5, Ϫ1, 4͒, and R͑0, 3, 8͒ is closest to the xz-plane? Which point lies in the yz-plane? 4. What are the projections of the point (2, 3, 5) on the xy-, yz-, 15–18 |||| Show that the equation represents a sphere, and find its center and radius. 15. x 2 ϩ y 2 ϩ z 2 Ϫ 6x ϩ 4y Ϫ 2z ෇ 11 16. x 2 ϩ y 2 ϩ z 2 ෇ 4x Ϫ 2y 17. x 2 ϩ y 2 ϩ z 2 ෇ x ϩ y ϩ z 18. 4x 2 ϩ 4y 2 ϩ 4z 2 Ϫ 8x ϩ 16y ෇ 1 ■ ■ ͩ and R͑Ϫ1, 1, 2͒ is an equilateral triangle. 8. Find the lengths of the sides of the triangle with vertices A͑1, 2, Ϫ3͒, B͑3, 4, Ϫ2͒, and C͑3, Ϫ2, 1͒. Is ABC a right triangle? Is it an isosceles triangle? 9. Determine whether the points lie on a straight line. (a) A͑5, 1, 3͒, B͑7, 9, Ϫ1͒, C͑1, Ϫ15, 11͒ (b) K͑0, 3, Ϫ4͒, L͑1, 2, Ϫ2͒, M͑3, 0, 1͒ 10. Find the distance from ͑3, 7, Ϫ5͒ to each of the following. (a) The xy-plane (c) The xz-plane (e) The y-axis (b) The yz-plane (d) The x-axis (f ) The z-axis 11. Find an equation of the sphere with center ͑1, Ϫ4, 3͒ and radius 5. What is the intersection of this sphere with the xz-plane? 12. Find an equation of the sphere with center ͑6, 5, Ϫ2͒ and radius s7. Describe its intersection with each of the coordinate planes. ■ ■ ■ ■ ■ ■ ■ x 1 ϩ x 2 y1 ϩ y2 z1 ϩ z2 , , 2 2 2 ͪ (b) Find the lengths of the medians of the triangle with vertices A͑1, 2, 3͒, B͑Ϫ2, 0, 5͒, and C͑4, 1, 5͒. tion x ϩ y ෇ 2. 7. Show that the triangle with vertices P͑Ϫ2, 4, 0͒, Q͑1, 2, Ϫ1͒, ■ P1͑x 1, y1, z1 ͒ to P2͑x 2 , y2 , z2 ͒ is 5. Describe and sketch the surface in ‫ 3ޒ‬represented by the equa- it represent in ‫ ? 3ޒ‬Illustrate with sketches. (b) What does the equation y ෇ 3 represent in ‫ ? 3ޒ‬What does z ෇ 5 represent? What does the pair of equations y ෇ 3, z ෇ 5 represent? In other words, describe the set of points ͑x, y, z͒ such that y ෇ 3 and z ෇ 5. Illustrate with a sketch. ■ 19. (a) Prove that the midpoint of the line segment from and xz-planes? Draw a rectangular box with the origin and ͑2, 3, 5͒ as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box. 6. (a) What does the equation x ෇ 4 represent in ‫ ? 2ޒ‬What does ■ 20. Find an equation of a sphere if one of its diameters has end- points ͑2, 1, 4͒ and ͑4, 3, 10͒. 21. Find equations of the spheres with center ͑2, Ϫ3, 6͒ that touch (a) the xy-plane, (b) the yz-plane, (c) the xz-plane. 22. Find an equation of the largest sphere with center (5, 4, 9) that is contained in the first octant. 3 |||| Describe in words the region of ‫ ޒ‬represented by the equation or inequality. 23–34 23. y ෇ Ϫ4 24. x ෇ 10 25. x Ͼ 3 26. y ജ 0 27. 0 ഛ z ഛ 6 28. y ෇ z 29. x 2 ϩ y 2 ϩ z 2 Ͼ 1 30. 1 ഛ x 2 ϩ y 2 ϩ z 2 ഛ 25 31. x 2 ϩ y 2 ϩ z 2 Ϫ 2z Ͻ 3 32. x 2 ϩ y 2 ෇ 1 33. x 2 ϩ z 2 ഛ 9 ■ ■ 35–38 ■ |||| 34. xyz ෇ 0 ■ ■ ■ ■ ■ ■ ■ ■ ■ Write inequalities to describe the region. 35. The half-space consisting of all points to the left of the xz-plane 36. The solid rectangular box in the first octant bounded by the planes x ෇ 1, y ෇ 2, and z ෇ 3 37. The region consisting of all points between (but not on) the spheres of radius r and R centered at the origin, where r Ͻ R 13. Find an equation of the sphere that passes through the point ͑4, 3, Ϫ1͒ and has center ͑3, 8, 1͒. 38. The solid upper hemisphere of the sphere of radius 2 centered at the origin 14. Find an equation of the sphere that passes through the origin and whose center is ͑1, 2, 3͒. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 5E-13(pp 828-837) 834 ❙❙❙❙ 1/18/06 11:10 AM Page 834 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE 39. The figure shows a line L 1 in space and a second line L 2 , which is the projection of L 1 on the xy-plane. (In other words, z L¡ the points on L 2 are directly beneath, or above, the points on L 1.) (a) Find the coordinates of the point P on the line L 1. (b) Locate on the diagram the points A, B, and C, where the line L1 intersects the xy-plane, the yz-plane, and the xz-plane, respectively. 40. Consider the points P such that the distance from P to P A͑Ϫ1, 5, 3͒ is twice the distance from P to B͑6, 2, Ϫ2͒. Show that the set of all such points is a sphere, and find its center and radius. 1 0 1 41. Find an equation of the set of all points equidistant from the L™ 1 points A͑Ϫ1, 5, 3͒ and B͑6, 2, Ϫ2͒. Describe the set. y x 42. Find the volume of the solid that lies inside both of the spheres x 2 ϩ y 2 ϩ z 2 ϩ 4x Ϫ 2y ϩ 4z ϩ 5 ෇ 0 and |||| 13.2 x 2 ϩ y 2 ϩ z2 ෇ 4 Vectors D B u v C A FIGURE 1 Equivalent vectors The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction. A vector is often represented by an arrow or a directed line segment. The length of the arrow represents the magnitude of the vector and the arrow points in the direction of the vector. We denote a vector by printing a l letter in boldface ͑v͒ or by putting an arrow above the letter ͑v ͒. For instance, suppose a particle moves along a line segment from point A to point B. The corresponding displacement vector v, shown in Figure 1, has initial point A (the tail) l and terminal point B (the tip) and we indicate this by writing v ෇ AB. Notice that the vecl tor u ෇ CD has the same length and the same direction as v even though it is in a different position. We say that u and v are equivalent (or equal) and we write u ෇ v. The zero vector, denoted by 0, has length 0. It is the only vector with no specific direction. Combining Vectors C B A FIGURE 2 l Suppose a particle moves from A to B, so its displacement vector is AB. Then the particle l changes direction and moves from B to C, with displacement vector BC as in Figure 2. The combined effect of these displacements is that the particle has moved from A to C. The l l l resulting displacement vector AC is called the sum of AB and BC and we write l l l AC ෇ AB ϩ BC In general, if we start with vectors u and v, we first move v so that its tail coincides with the tip of u and define the sum of u and v as follows. Definition of Vector Addition If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u ϩ v is the vector from the initial point of u to the terminal point of v. 5E-13(pp 828-837) 1/18/06 11:10 AM Page 835 SECTION 13.2 VECTORS ❙❙❙❙ 835 The definition of vector addition is illustrated in Figure 3. You can see why this definition is sometimes called the Triangle Law. u u+v v v u v v+ u+ v u u FIGURE 4 The Parallelogram Law FIGURE 3 The Triangle Law In Figure 4 we start with the same vectors u and v as in Figure 3 and draw another copy of v with the same initial point as u. Completing the parallelogram, we see that u ϩ v ෇ v ϩ u. This also gives another way to construct the sum: If we place u and v so they start at the same point, then u ϩ v lies along the diagonal of the parallelogram with u and v as sides. (This is called the Parallelogram Law.) EXAMPLE 1 Draw the sum of the vectors a and b shown in Figure 5. a b SOLUTION First we translate b and place its tail at the tip of a, being careful to draw a copy of b that has the same length and direction. Then we draw the vector a ϩ b [see Figure 6(a)] starting at the initial point of a and ending at the terminal point of the copy of b. Alternatively, we could place b so it starts where a starts and construct a ϩ b by the Parallelogram Law as in Figure 6(b). FIGURE 5 a Visual 13.2 shows how the Triangle and Parallelogram Laws work for various vectors u and v. FIGURE 6 a b a+b a+b b (a) (b) It is possible to multiply a vector by a real number c. (In this context we call the real number c a scalar to distinguish it from a vector.) For instance, we want 2v to be the same vector as v ϩ v, which has the same direction as v but is twice as long. In general, we multiply a vector by a scalar as follows. Definition of Scalar Multiplication If c is a scalar and v is a vector, then the scalar mul- Խ Խ 2v v _v _1.5v FIGURE 7 Scalar multiples of v 1 2v tiple cv is the vector whose length is c times the length of v and whose direction is the same as v if c Ͼ 0 and is opposite to v if c Ͻ 0. If c ෇ 0 or v ෇ 0, then cv ෇ 0. This definition is illustrated in Figure 7. We see that real numbers work like scaling factors here; that’s why we call them scalars. Notice that two nonzero vectors are parallel if they are scalar multiples of one another. In particular, the vector Ϫv ෇ ͑Ϫ1͒v has the same length as v but points in the opposite direction. We call it the negative of v. By the difference u Ϫ v of two vectors we mean u Ϫ v ෇ u ϩ ͑Ϫv͒ 5E-13(pp 828-837) 836 ❙❙❙❙ 1/18/06 11:11 AM Page 836 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE So we can construct u Ϫ v by first drawing the negative of v, Ϫv, and then adding it to u by the Parallelogram Law as in Figure 8(a). Alternatively, since v ϩ ͑u Ϫ v͒ ෇ u, the vector u Ϫ v, when added to v, gives u. So we could construct u Ϫ v as in Figure 8(b) by means of the Triangle Law. v u u-v u-v _v v u FIGURE 8 Drawing u-v (a) (b) EXAMPLE 2 If a and b are the vectors shown in Figure 9, draw a Ϫ 2b. SOLUTION We first draw the vector Ϫ2b pointing in the direction opposite to b and twice as long. We place it with its tail at the tip of a and then use the Triangle Law to draw a ϩ ͑Ϫ2b͒ as in Figure 10. a _2b a b a-2b FIGURE 9 FIGURE 10 Components y For some purposes it’s best to introduce a coordinate system and treat vectors algebraically. If we place the initial point of a vector a at the origin of a rectangular coordinate system, then the terminal point of a has coordinates of the form ͑a1, a2 ͒ or ͑a1, a2, a3͒, depending on whether our coordinate system is two- or three-dimensional (see Figure 11). These coordinates are called the components of a and we write (a¡, a™) a O x a ෇ ͗a 1, a 2 ͘ a=ka¡, a™l z (a¡, a™, a£) a O y x or a ෇ ͗a 1, a 2 , a 3 ͘ We use the notation ͗a1, a2 ͘ for the ordered pair that refers to a vector so as not to confuse it with the ordered pair ͑a1, a2 ͒ that refers to a point in the plane. For instance, the vectors shown in Figure 12 are all equivalent to the vector l OP ෇ ͗3, 2͘ whose terminal point is P͑3, 2͒. What they have in common is that the terminal point is reached from the initial point by a displacement of three units to the right and two upward. We can think of all these geometric vectors as representations of the y a=ka¡, a™, a£l z (4, 5) FIGURE 11 (1, 3) position vector of P P(3, 2) P(a¡, a™, a£) 0 x O x FIGURE 12 Representations of the vector a=k3, 2l B(x+a¡, y+a™, z+a£) A(x, y, z) FIGURE 13 Representations of a=ka¡, a™, a£l y 5E-13(pp 828-837) 1/18/06 11:11 AM Page 837 SECTION 13.2 VECTORS ❙❙❙❙ 837 l algebraic vector a ෇ ͗3, 2͘ . The particular representation OP from the origin to the point P͑3, 2͒ is called the position vector of the point P. l In three dimensions, the vector a ෇ OP ෇ ͗a1, a2, a3 ͘ is the position vector of the l point P͑a1, a2, a3͒. (See Figure 13.) Let’s consider any other representation AB of a, where the initial point is A͑x 1, y1, z1 ͒ and the terminal point is B͑x 2 , y2 , z2 ͒. Then we must have x 1 ϩ a 1 ෇ x 2, y1 ϩ a 2 ෇ y2, and z1 ϩ a 3 ෇ z2 and so a 1 ෇ x 2 Ϫ x 1, a 2 ෇ y2 Ϫ y1, and a 3 ෇ z2 Ϫ z1. Thus, we have the following result. 1 Given the points A͑x 1, y1, z1 ͒ and B͑x 2 , y2 , z2 ͒, the vector a with represenl tation AB is a ෇ ͗x 2 Ϫ x 1, y2 Ϫ y1, z2 Ϫ z1 ͘ EXAMPLE 3 Find the vector represented by the directed line segment with initial point A͑2, Ϫ3, 4) and terminal point B͑Ϫ2, 1, 1͒. l SOLUTION By (1), the vector corresponding to AB is a ෇ ͗Ϫ2 Ϫ 2, 1 Ϫ ͑Ϫ3͒, 1 Ϫ 4͘ ෇ ͗Ϫ4, 4, Ϫ3͘ The magnitude or length of the vector v is the length of any of its representations and is denoted by the symbol v or ʈ v ʈ. By using the distance formula to compute the length of a segment OP, we obtain the following formulas. Խ Խ The length of the two-dimensional vector a ෇ ͗a 1, a 2 ͘ is Խ a Խ ෇ sa 2 1 ϩ a2 2 The length of the three-dimensional vector a ෇ ͗a 1, a 2 , a 3 ͘ is y (a¡+b¡, a™+b™) a+b b¡ a 0 a™ a™ a¡ 2 1 ϩ a2 ϩ a2 2 3 How do we add vectors algebraically? Figure 14 shows that if a ෇ ͗a 1, a 2 ͘ and b ෇ ͗b 1, b 2 ͘ , then the sum is a ϩ b ෇ ͗a1 ϩ b1, a2 ϩ b2 ͘ , at least for the case where the components are positive. In other words, to add algebraic vectors we add their components. Similarly, to subtract vectors we subtract components. From the similar triangles in Figure 15 we see that the components of ca are ca1 and ca2. So to multiply a vector by a scalar we multiply each component by that scalar. b™ b Խ a Խ ෇ sa x b¡ If a ෇ ͗a 1, a 2 ͘ and b ෇ ͗b1, b2 ͘ , then FIGURE 14 a ϩ b ෇ ͗a 1 ϩ b1, a 2 ϩ b2 ͘ a Ϫ b ෇ ͗a 1 Ϫ b1, a 2 Ϫ b2 ͘ ca ෇ ͗ca1, ca2 ͘ Similarly, for three-dimensional vectors, ca a ca™ a™ ͗a 1, a 2 , a 3 ͘ ϩ ͗b1, b2 , b3 ͘ ෇ ͗a 1 ϩ b1, a 2 ϩ b2 , a 3 ϩ b3 ͘ ͗a 1, a 2 , a 3 ͘ Ϫ ͗b1, b2 , b3 ͘ ෇ ͗a 1 Ϫ b1, a 2 Ϫ b2 , a 3 Ϫ b3 ͘ a¡ FIGURE 15 ca¡ c͗a 1, a 2 , a 3 ͘ ෇ ͗ca1, ca2 , ca3 ͘ 5E-13(pp 838-847) 838 ❙❙❙❙ 1/18/06 11:13 AM Page 838 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE Խ Խ EXAMPLE 4 If a ෇ ͗4, 0, 3͘ and b ෇ ͗Ϫ2, 1, 5͘ , find a and the vectors a ϩ b, a Ϫ b, 3b, and 2a ϩ 5b. Խ a Խ ෇ s4 SOLUTION 2 ϩ 0 2 ϩ 32 ෇ s25 ෇ 5 a ϩ b ෇ ͗4, 0, 3͘ ϩ ͗Ϫ2, 1, 5͘ ෇ ͗4 Ϫ 2, 0 ϩ 1, 3 ϩ 5͘ ෇ ͗2, 1, 8 ͘ a Ϫ b ෇ ͗4, 0, 3͘ Ϫ ͗Ϫ2, 1, 5͘ ෇ ͗4 Ϫ ͑Ϫ2͒, 0 Ϫ 1, 3 Ϫ 5͘ ෇ ͗6, Ϫ1, Ϫ2͘ 3b ෇ 3͗Ϫ2, 1, 5͘ ෇ ͗3͑Ϫ2͒, 3͑1͒, 3͑5͒͘ ෇ ͗Ϫ6, 3, 15 ͘ 2a ϩ 5b ෇ 2͗4, 0, 3͘ ϩ 5͗Ϫ2, 1, 5͘ ෇ ͗8, 0, 6͘ ϩ ͗Ϫ10, 5, 25͘ ෇ ͗Ϫ2, 5, 31͘ We denote by V2 the set of all two-dimensional vectors and by V3 the set of all threedimensional vectors. More generally, we will later need to consider the set Vn of all n-dimensional vectors. An n-dimensional vector is an ordered n-tuple: |||| Vectors in n dimensions are used to list various quantities in an organized way. For instance, the components of a six-dimensional vector p ෇ ͗ p1 , p2 , p3 , p4 , p5 , p6 ͘ might represent the prices of six different ingredients required to make a particular product. Four-dimensional vectors ͗ x, y, z, t͘ are used in relativity theory, where the first three components specify a position in space and the fourth represents time. a ෇ ͗a1, a 2, . . . , a n ͘ where a1, a 2, . . . , a n are real numbers that are called the components of a. Addition and scalar multiplication are defined in terms of components just as for the cases n ෇ 2 and n ෇ 3. Properties of Vectors If a, b, and c are vectors in Vn and c and d are scalars, then 1. a ϩ b ෇ b ϩ a 2. a ϩ ͑b ϩ c͒ ෇ ͑a ϩ b͒ ϩ c 3. a ϩ 0 ෇ a 4. a ϩ ͑Ϫa͒ ෇ 0 5. c͑a ϩ b͒ ෇ ca ϩ cb 6. ͑c ϩ d͒a ෇ ca ϩ da 7. ͑cd ͒a ෇ c͑da͒ 8. 1a ෇ a These eight properties of vectors can be readily verified either geometrically or algebraically. For instance, Property 1 can be seen from Figure 4 (it’s equivalent to the Parallelogram Law) or as follows for the case n ෇ 2: a ϩ b ෇ ͗a 1, a 2 ͘ ϩ ͗b1, b2 ͘ ෇ ͗a 1 ϩ b1, a 2 ϩ b2 ͘ ෇ ͗b1 ϩ a 1, b2 ϩ a 2 ͘ ෇ ͗b1, b2 ͘ ϩ ͗a 1, a 2 ͘ Q ෇bϩa c (a+b)+c =a+(b+c) b a+b b+c P FIGURE 16 We can see why Property 2 (the associative law) is true by looking at Figure 16 and l applying the Triangle Law several times: The vector PQ is obtained either by first constructing a ϩ b and then adding c or by adding a to the vector b ϩ c. Three vectors in V3 play a special role. Let a i ෇ ͗1, 0, 0͘ j ෇ ͗0, 1, 0͘ k ෇ ͗0, 0, 1͘ 5E-13(pp 838-847) 1/18/06 11:13 AM Page 839 SECTION 13.2 VECTORS ❙❙❙❙ 839 Then i , j, and k are vectors that have length 1 and point in the directions of the positive x-, y-, and z-axes. Similarly, in two dimensions we define i ෇ ͗1, 0͘ and j ෇ ͗0, 1͘ . (See Figure 17.) y z j k (0, 1) 0 x i i (1, 0) FIGURE 17 Standard basis vectors in V™ and V£ j y x (a) (b) If a ෇ ͗a 1, a 2 , a 3 ͘ , then we can write a ෇ ͗a 1, a 2 , a 3 ͘ ෇ ͗a 1, 0, 0͘ ϩ ͗0, a 2 , 0͘ ϩ ͗0, 0, a 3 ͘ ෇ a 1 ͗1, 0, 0͘ ϩ a 2 ͗0, 1, 0͘ ϩ a 3 ͗0, 0, 1 ͘ y (a¡, a™) a a™ j a¡i 0 2 a ෇ a1 i ϩ a2 j ϩ a3 k Thus, any vector in V3 can be expressed in terms of the standard basis vectors i , j, and k. For instance, x ͗1, Ϫ2, 6͘ ෇ i Ϫ 2j ϩ 6k (a) a=a¡i+a™ j Similarly, in two dimensions, we can write z 3 a ෇ ͗a1, a2 ͘ ෇ a1 i ϩ a2 j (a¡, a™, a£) See Figure 18 for the geometric interpretation of Equations 3 and 2 and compare with Figure 17. a a£k a¡i y x a™ j (b) a=a¡i+a™ j+a£k EXAMPLE 5 If a ෇ i ϩ 2j Ϫ 3k and b ෇ 4i ϩ 7 k, express the vector 2a ϩ 3b in terms of i , j, and k. SOLUTION Using Properties 1, 2, 5, 6, and 7 of vectors, we have FIGURE 18 2a ϩ 3b ෇ 2͑i ϩ 2j Ϫ 3k͒ ϩ 3͑4i ϩ 7k͒ ෇ 2i ϩ 4j Ϫ 6k ϩ 12i ϩ 21k ෇ 14i ϩ 4j ϩ 15k A unit vector is a vector whose length is 1. For instance, i , j, and k are all unit vectors. In general, if a 0, then the unit vector that has the same direction as a is 4 u෇ 1 a a෇ a a Խ Խ Խ Խ Խ Խ In order to verify this, we let c ෇ 1͞ a . Then u ෇ ca and c is a positive scalar, so u has the same direction as a. Also 1 Խ u Խ ෇ Խ ca Խ ෇ Խ c ԽԽ a Խ ෇ Խ a Խ Խ a Խ ෇ 1 5E-13(pp 838-847) 840 ❙❙❙❙ 1/18/06 11:14 AM Page 840 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 6 Find the unit vector in the direction of the vector 2i Ϫ j Ϫ 2k. SOLUTION The given vector has length Խ 2i Ϫ j Ϫ 2k Խ ෇ s2 2 ϩ ͑Ϫ1͒2 ϩ ͑Ϫ2͒2 ෇ s9 ෇ 3 so, by Equation 4, the unit vector with the same direction is 1 3 ͑2i Ϫ j Ϫ 2k͒ ෇ 2 i Ϫ 1 j Ϫ 2 k 3 3 3 Applications Vectors are useful in many aspects of physics and engineering. In Chapter 14 we will see how they describe the velocity and acceleration of objects moving in space. Here we look at forces. A force is represented by a vector because it has both a magnitude (measured in pounds or newtons) and a direction. If several forces are acting on an object, the resultant force experienced by the object is the vector sum of these forces. 50° 32° T¡ T™ EXAMPLE 7 A 100-lb weight hangs from two wires as shown in Figure 19. Find the tensions (forces) T1 and T2 in both wires and their magnitudes. SOLUTION We first express T1 and T2 in terms of their horizontal and vertical components. From Figure 20 we see that 100 Խ Խ Խ Խ ෇ Խ T Խ cos 32Њ i ϩ Խ T Խ sin 32Њ j 5 6 FIGURE 19 T1 ෇ Ϫ T1 cos 50Њ i ϩ T1 sin 50Њ j T2 2 2 The resultant T1 ϩ T2 of the tensions counterbalances the weight w and so we must have 50° T¡ T™ 50° 32° 32° T1 ϩ T2 ෇ Ϫw ෇ 100j Thus (ϪԽ T1 Խ cos 50Њ ϩ Խ T2 Խ cos 32Њ) i ϩ (Խ T1 Խ sin 50Њ ϩ Խ T2 Խ sin 32Њ) j ෇ 100j w Equating components, we get FIGURE 20 Խ Խ Խ Խ Խ T Խ sin 50Њ ϩ Խ T Խ sin 32Њ ෇ 100 Solving the first of these equations for Խ T Խ and substituting into the second, we get T cos 50Њ sin 32Њ ෇ 100 Խ T Խ sin 50Њ ϩ Խ Խ Ϫ T1 cos 50Њ ϩ T2 cos 32Њ ෇ 0 1 2 2 1 1 cos 32Њ So the magnitudes of the tensions are ԽT Խ ෇ 1 and 100 Ϸ 85.64 lb sin 50Њ ϩ tan 32Њ cos 50Њ T Խ cos 50Њ Խ T Խ ෇ Խ cos 32Њ Ϸ 64.91 lb 1 2 Substituting these values in (5) and (6), we obtain the tension vectors T1 Ϸ Ϫ55.05 i ϩ 65.60 j T2 Ϸ 55.05 i ϩ 34.40 j 5E-13(pp 838-847) 1/18/06 11:14 AM Page 841 SECTION 13.2 VECTORS |||| 13.2 841 Exercises 9. A͑Ϫ1, Ϫ1͒, 1. Are the following quantities vectors or scalars? Explain. (a) (b) (c) (d) ❙❙❙❙ The cost of a theater ticket The current in a river The initial flight path from Houston to Dallas The population of the world 11. A͑0, 3, 1͒, ■ 2. What is the relationship between the point (4, 7) and the vector ͗4, 7 ͘ ? Illustrate with a sketch. 3. Name all the equal vectors in the parallelogram shown. A B ■ B͑2, 3, Ϫ1͒ ■ ■ ■ 13. ͗3, Ϫ1 ͘ , 12. A͑4, 0, Ϫ2͒, ■ 15. ͗0, 1, 2 ͘ , ͗0, 0, Ϫ3͘ ■ ■ |||| ■ l l (a) PQ ϩ QR l l (c) QS Ϫ PS ■ ■ ■ ■ ■ ͗5, 7 ͘ ͗0, 4, 0͘ ■ ■ ■ ■ ■ ■ b ෇ ͗6, 2͘ b ෇ i ϩ 5j b ෇ ͗Ϫ1, 5, Ϫ2 ͘ 20. a ෇ ͗Ϫ3, Ϫ4, Ϫ1͘ , 22. a ෇ 3 i Ϫ 2 k, ■ P ■ Խ Խ 21. a ෇ i Ϫ 2 j ϩ k, Q B͑4, 2, 1͒ ■ Find a , a ϩ b, a Ϫ b, 2a, and 3a ϩ 4b. 19. a ෇ ͗6, 2, 3͘ , l l (b) RP ϩ PS l l l (d) RS ϩ SP ϩ PQ ■ 16. ͗Ϫ1, 0, 2͘ , ■ 18. a ෇ 2 i Ϫ 3 j, 4. Write each combination of vectors as a single vector. ■ 14. ͗Ϫ2, Ϫ1͘ , 17. a ෇ ͗Ϫ4, 3 ͘ , C ■ ͗Ϫ2, 4 ͘ 17–22 D B͑3, 0͒ 13–16 |||| Find the sum of the given vectors and illustrate geometrically. ■ E 10. A͑Ϫ2, 2͒, B͑Ϫ3, 4͒ ■ 23–25 ■ |||| b ෇ ͗6, 2, Ϫ3 ͘ b ෇ j ϩ 2k b෇iϪjϩk ■ ■ ■ ■ ■ ■ Find a unit vector that has the same direction as the given vector. 23. ͗9, Ϫ5͘ S R 25. 8 i Ϫ j ϩ 4 k 5. Copy the vectors in the figure and use them to draw the following vectors. (a) u ϩ v (c) v ϩ w 24. 12 i Ϫ 5 j ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 26. Find a vector that has the same direction as ͗Ϫ2, 4, 2͘ but has (b) u Ϫ v (d) w ϩ v ϩ u length 6. 27. If v lies in the first quadrant and makes an angle ␲͞3 with the u v Խ Խ positive x-axis and v ෇ 4, find v in component form. w 28. If a child pulls a sled through the snow with a force of 50 N 6. Copy the vectors in the figure and use them to draw the follow- ing vectors. (a) a ϩ b (c) 2a (e) 2a ϩ b (b) a Ϫ b (d) Ϫ 1 b 2 (f) b Ϫ 3a a exerted at an angle of 38Њ above the horizontal, find the horizontal and vertical components of the force. 29. Two forces F1 and F2 with magnitudes 10 lb and 12 lb act on an object at a point P as shown in the figure. Find the resultant force F acting at P as well as its magnitude and its direction. (Indicate the direction by finding the angle ␪ shown in the figure.) F b F¡ 7–12 Find a vector a with representation given by the directed l l line segment AB. Draw AB and the equivalent representation starting at the origin. F™ |||| 7. A͑2, 3͒, B͑Ϫ2, 1͒ 8. A͑Ϫ2, Ϫ2͒, ■ B͑5, 3͒ ¨ 30° 45° P 5E-13(pp 838-847) 842 ❙❙❙❙ 1/18/06 11:14 AM Page 842 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE 37. (a) Draw the vectors a ෇ ͗3, 2͘ , b ෇ ͗2, Ϫ1 ͘ , and c ෇ ͗7, 1͘. 30. Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N45Њ W at a speed of 50 km͞h. (This means that the direction from which the wind blows is 45Њ west of the northerly direction.) A pilot is steering a plane in the direction N60Њ E at an airspeed (speed in still air) of 250 km͞h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane. 31. A woman walks due west on the deck of a ship at 3 mi͞h. The ship is moving north at a speed of 22 mi͞h. Find the speed and direction of the woman relative to the surface of the water. 32. Ropes 3 m and 5 m in length are fastened to a holiday decora- tion that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, make angles of 52Њ and 40Њ with the horizontal. Find the tension in each wire and the magnitude of each tension. 52° 3 m 40° (b) Show, by means of a sketch, that there are scalars s and t such that c ෇ sa ϩ t b. (c) Use the sketch to estimate the values of s and t. (d) Find the exact values of s and t. 38. Suppose that a and b are nonzero vectors that are not parallel and c is any vector in the plane determined by a and b. Give a geometric argument to show that c can be written as c ෇ sa ϩ t b for suitable scalars s and t. Then give an argument using components. 39. If r ෇ ͗x, y, z͘ and r0 ෇ ͗x 0 , y0 , z0 ͘ , describe the set of all Խ Խ points ͑x, y, z͒ such that r Ϫ r0 ෇ 1. 40. If r ෇ ͗x, y͘ , r1 ෇ ͗x 1, y1 ͘ , and r2 ෇ ͗x 2 , y2 ͘ , describe the Խ Խ Խ Խ set of all points ͑x, y͒ such that r Ϫ r1 ϩ r Ϫ r2 ෇ k, where k Ͼ r1 Ϫ r2 . Խ Խ 41. Figure 16 gives a geometric demonstration of Property 2 of vectors. Use components to give an algebraic proof of this fact for the case n ෇ 2. 42. Prove Property 5 of vectors algebraically for the case n ෇ 3. Then use similar triangles to give a geometric proof. 5 m 43. Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. 44. Suppose the three coordinate planes are all mirrored and a 33. A clothesline is tied between two poles, 8 m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the midpoint is pulled down 8 cm. Find the tension in each half of the clothesline. 34. The tension T at each end of the chain has magnitude 25 N. What is the weight of the chain? 37° light ray given by the vector a ෇ ͗a 1, a 2 , a 3 ͘ first strikes the xz-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by b ෇ ͗a 1, Ϫa 2 , a 3 ͘ . Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the Moon, to calculate very precisely the distance from the Earth to the Moon.) z 37° 35. If A, B, and C are the vertices of a triangle, find l l l AB ϩ BC ϩ CA. b 36. Let C be the point on the line segment AB that is twice as far l l l from B as it is from A. If a ෇ OA, b ෇ OB, and c ෇ OC, show 2 1 that c ෇ 3 a ϩ 3 b. a x y 5E-13(pp 838-847) 1/18/06 11:15 AM Page 843 SECTION 13.3 THE DOT PRODUCT |||| 13.3 ❙❙❙❙ 843 The Dot Product So far we have added two vectors and multiplied a vector by a scalar. The question arises: Is it possible to multiply two vectors so that their product is a useful quantity? One such product is the dot product, whose definition follows. Another is the cross product, which is discussed in the next section. 1 Definition If a ෇ ͗a 1, a 2 , a 3 ͘ and b ෇ ͗b1, b2 , b3 ͘ , then the dot product of a and b is the number a ؒ b given by a ؒ b ෇ a 1 b1 ϩ a 2 b2 ϩ a 3 b3 Thus, to find the dot product of a and b we multiply corresponding components and add. The result is not a vector. It is a real number, that is, a scalar. For this reason, the dot product is sometimes called the scalar product (or inner product). Although Definition 1 is given for three-dimensional vectors, the dot product of two-dimensional vectors is defined in a similar fashion: ͗a 1, a 2 ͘ ؒ ͗b1, b2 ͘ ෇ a 1 b1 ϩ a 2 b2 EXAMPLE 1 ͗2, 4͘ ؒ ͗3, Ϫ1͘ ෇ 2͑3͒ ϩ 4͑Ϫ1͒ ෇ 2 ͗Ϫ1, 7, 4͘ ؒ ͗6, 2, Ϫ 1 ͘ ෇ ͑Ϫ1͒͑6͒ ϩ 7͑2͒ ϩ 4(Ϫ 1 ) ෇ 6 2 2 ͑i ϩ 2 j Ϫ 3k͒ ؒ ͑2 j Ϫ k͒ ෇ 1͑0͒ ϩ 2͑2͒ ϩ ͑Ϫ3͒͑Ϫ1͒ ෇ 7 The dot product obeys many of the laws that hold for ordinary products of real numbers. These are stated in the following theorem. 2 Properties of the Dot Product If a, b, and c are vectors in V3 and c is a scalar, then Խ Խ 1. a ؒ a ෇ a 2. a ؒ b ෇ b ؒ a 2 3. a ؒ ͑b ϩ c͒ ෇ a ؒ b ϩ a ؒ c 4. ͑ca͒ ؒ b ෇ c͑a ؒ b͒ ෇ a ؒ ͑cb͒ 5. 0 ؒ a ෇ 0 These properties are easily proved using Definition 1. For instance, here are the proofs of Properties 1 and 3: Խ Խ 1. a ؒ a ෇ a 2 ϩ a 2 ϩ a 2 ෇ a 1 2 3 2 3. a ؒ ͑b ϩ c͒ ෇ ͗a1, a2, a3 ͘ ؒ ͗b1 ϩ c1, b2 ϩ c2, b3 ϩ c3 ͘ ෇ a 1͑b1 ϩ c1͒ ϩ a 2͑b2 ϩ c2͒ ϩ a 3͑b3 ϩ c3͒ ෇ a 1 b1 ϩ a 1 c1 ϩ a 2 b2 ϩ a 2 c2 ϩ a 3 b3 ϩ a 3 c3 ෇ ͑a 1 b1 ϩ a 2 b2 ϩ a 3 b3͒ ϩ ͑a 1 c1 ϩ a 2 c2 ϩ a 3 c3 ͒ ෇aؒbϩaؒc The proofs of the remaining properties are left as exercises. The dot product a ؒ b can be given a geometric interpretation in terms of the angle ␪ between a and b, which is defined to be the angle between the representations of a and b that start at the origin, where 0 ഛ ␪ ഛ ␲. In other words, ␪ is the angle between the 5E-13(pp 838-847) 844 ❙❙❙❙ 1/18/06 11:15 AM Page 844 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE l l line segments OA and OB in Figure 1. Note that if a and b are parallel vectors, then ␪ ෇ 0 or ␪ ෇ ␲. The formula in the following theorem is used by physicists as the definition of the dot product. z B a-b b 0 ¨ x a A 3 Theorem If ␪ is the angle between the vectors a and b, then Խ ԽԽ b Խ cos ␪ aؒb෇ a y FIGURE 1 Proof If we apply the Law of Cosines to triangle OAB in Figure 1, we get Խ AB Խ 2 4 Խ ෇ OA Խ 2 Խ ϩ OB Խ 2 Խ Ϫ 2 OA ԽԽ OB Խ cos ␪ (Observe that the Law of Cosines still applies in the limiting cases when ␪ ෇ 0 or ␲, or a ෇ 0 or b ෇ 0.) But OA ෇ a , OB ෇ b , and AB ෇ a Ϫ b , so Equation 4 becomes Խ Խ Խ ԽԽ Խa Ϫ bԽ 2 5 Խ Խ Խ Խ Խ ෇ a 2 Խ Խ ϩ b 2 Խ Խ Խ Խ Խ ԽԽ b Խ cos ␪ Ϫ2 a Using Properties 1, 2, and 3 of the dot product, we can rewrite the left side of this equation as follows: a Ϫ b 2 ෇ ͑a Ϫ b͒ ؒ ͑a Ϫ b͒ Խ Խ ෇aؒaϪaؒbϪbؒaϩbؒb Խ Խ ෇ a 2 Խ Խ Ϫ 2a ؒ b ϩ b 2 Therefore, Equation 5 gives ԽaԽ 2 Thus or Խ Խ Խ Խ Ϫ 2 Խ a ԽԽ b Խ cos ␪ Ϫ2a ؒ b ෇ Ϫ2 Խ a ԽԽ b Խ cos ␪ a ؒ b ෇ Խ a ԽԽ b Խ cos ␪ Ϫ 2a ؒ b ϩ b 2 Խ Խ ෇ a 2 ϩ b 2 EXAMPLE 2 If the vectors a and b have lengths 4 and 6, and the angle between them is ␲͞3, find a ؒ b. SOLUTION Using Theorem 3, we have Խ ԽԽ b Խ cos͑␲͞3͒ ෇ 4 ؒ 6 ؒ aؒb෇ a 1 2 ෇ 12 The formula in Theorem 3 also enables us to find the angle between two vectors. 6 Corollary If ␪ is the angle between the nonzero vectors a and b, then cos ␪ ෇ aؒb a b Խ ԽԽ Խ EXAMPLE 3 Find the angle between the vectors a ෇ ͗2, 2, Ϫ1͘ and b ෇ ͗5, Ϫ3, 2͘ . SOLUTION Since Խ a Խ ෇ s2 2 ϩ 2 2 ϩ ͑Ϫ1͒2 ෇ 3 and Խ b Խ ෇ s5 2 ϩ ͑Ϫ3͒2 ϩ 2 2 ෇ s38 5E-13(pp 838-847) 1/18/06 11:16 AM Page 845 SECTION 13.3 THE DOT PRODUCT ❙❙❙❙ 845 and since a ؒ b ෇ 2͑5͒ ϩ 2͑Ϫ3͒ ϩ ͑Ϫ1͒͑2͒ ෇ 2 we have, from Corollary 6, cos ␪ ෇ So the angle between a and b is aؒb 2 ෇ a b 3s38 Խ ԽԽ Խ ͩ ͪ ␪ ෇ cosϪ1 2 3s38 Ϸ 1.46 ͑or 84Њ͒ Two nonzero vectors a and b are called perpendicular or orthogonal if the angle between them is ␪ ෇ ␲͞2. Then Theorem 3 gives Խ ԽԽ b Խ cos͑␲͞2͒ ෇ 0 aؒb෇ a and conversely if a ؒ b ෇ 0, then cos ␪ ෇ 0, so ␪ ෇ ␲͞2. The zero vector 0 is considered to be perpendicular to all vectors. Therefore, we have the following method for determining whether two vectors are orthogonal. a and b are orthogonal if and only if a ؒ b ෇ 0. 7 EXAMPLE 4 Show that 2i ϩ 2j Ϫ k is perpendicular to 5i Ϫ 4j ϩ 2k. SOLUTION Since ͑2i ϩ 2j Ϫ k͒ ؒ ͑5i Ϫ 4j ϩ 2k͒ ෇ 2͑5͒ ϩ 2͑Ϫ4͒ ϩ ͑Ϫ1͒͑2͒ ෇ 0 a ¨ a b b b ¨ a · b>0 a · b =0 a · b<0 a these vectors are perpendicular by (7). Because cos ␪ Ͼ 0 if 0 ഛ ␪ Ͻ ␲͞2 and cos ␪ Ͻ 0 if ␲͞2 Ͻ ␪ ഛ ␲, we see that a ؒ b is positive for ␪ Ͻ ␲͞2 and negative for ␪ Ͼ ␲͞2. We can think of a ؒ b as measuring the extent to which a and b point in the same direction. The dot product a ؒ b is positive if a and b point in the same general direction, 0 if they are perpendicular, and negative if they point in generally opposite directions (see Figure 2). In the extreme case where a and b point in exactly the same direction, we have ␪ ෇ 0, so cos ␪ ෇ 1 and Խ ԽԽ b Խ FIGURE 2 Visual 13.3A shows an animation of Figure 2. aؒb෇ a If a and b point in exactly opposite directions, then ␪ ෇ ␲ and so cos ␪ ෇ Ϫ1 and aؒb෇Ϫ a b . Խ ԽԽ Խ Direction Angles and Direction Cosines The direction angles of a nonzero vector a are the angles ␣, ␤, and ␥ (in the interval ͓0, ␲͔͒ that a makes with the positive x-, y-, and z-axes (see Figure 3 on page 846). The cosines of these direction angles, cos ␣, cos ␤, and cos ␥, are called the direction cosines of the vector a. Using Corollary 6 with b replaced by i , we obtain 8 cos ␣ ෇ aؒi a1 ෇ a i a Խ ԽԽ Խ Խ Խ 5E-13(pp 838-847) 846 ❙❙❙❙ 1/18/06 11:16 AM Page 846 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z (This can also be seen directly from Figure 3.) Similarly, we also have cos ␤ ෇ 9 ç a¡ a ∫ a2 a cos ␥ ෇ Խ Խ a3 a Խ Խ By squaring the expressions in Equations 8 and 9 and adding, we see that å y cos 2␣ ϩ cos 2␤ ϩ cos 2␥ ෇ 1 10 x We can also use Equations 8 and 9 to write FIGURE 3 Խ Խ Խ Խ Խ Խ a ෇ ͗a 1, a 2 , a 3 ͘ ෇ ͗ a cos ␣, a cos ␤, a cos ␥͘ Խ Խ ෇ a ͗cos ␣, cos ␤, cos ␥ ͘ Therefore 1 a ෇ ͗ cos ␣, cos ␤, cos ␥ ͘ a 11 Խ Խ which says that the direction cosines of a are the components of the unit vector in the direction of a. EXAMPLE 5 Find the direction angles of the vector a ෇ ͗ 1, 2, 3͘ . Խ Խ SOLUTION Since a ෇ s1 2 ϩ 2 2 ϩ 3 2 ෇ s14, Equations 8 and 9 give cos ␣ ෇ and so ͩ ͪ ␣ ෇ cosϪ1 1 s14 1 s14 cos ␤ ෇ 2 s14 ͩ ͪ ␤ ෇ cosϪ1 Ϸ 74Њ 2 s14 cos ␥ ෇ 3 s14 ͩ ͪ ␥ ෇ cosϪ1 Ϸ 58Њ 3 s14 Ϸ 37Њ Projections l l Figure 4 shows representations PQ and PR of two vectors a and b with the same initial l point P. If S is the foot of the perpendicular from R to the line containing PQ, then the l vector with representation PS is called the vector projection of b onto a and is denoted by proja b. R Visual 13.3B shows how Figure 4 changes when we vary a and b. R b b a a FIGURE 4 Vector projections P S proj a b Q S P Q proj a b The scalar projection of b onto a (also called the component of b along a) is defined to be the magnitude of the vector projection, which is the number b cos ␪, where ␪ is the Խ Խ 5E-13(pp 838-847) 1/18/06 11:16 AM Page 847 SECTION 13.3 THE DOT PRODUCT b P 847 angle between a and b. (See Figure 5; you can think of the scalar projection of b as being the length of a shadow of b.) This is denoted by compa b. Observe that it is negative if ␲͞2 Ͻ ␪ ഛ ␲. The equation R a ¨ ❙❙❙❙ Խ ԽԽ b Խ cos ␪ ෇ Խ a Խ( Խ b Խ cos ␪ ) aؒb෇ a Q S ͉b͉ cos ¨ shows that the dot product of a and b can be interpreted as the length of a times the scalar projection of b onto a. Since FIGURE 5 Scalar projection aؒb Խ b Խ cos ␪ ෇ Խ a Խ ෇ a ؒb a Խ Խ the component of b along a can be computed by taking the dot product of b with the unit vector in the direction of a. We summarize these ideas as follows. Scalar projection of b onto a: compa b ෇ Vector projection of b onto a: proja b ෇ aؒb a Խ Խ ͩԽ ԽͪԽ Խ aؒb a a aؒb ෇ a a a 2 Խ Խ Notice that the vector projection is the scalar projection times the unit vector in the direction of a. EXAMPLE 6 Find the scalar projection and vector projection of b ෇ ͗1, 1, 2 ͘ onto a ෇ ͗Ϫ2, 3, 1͘ . Խ Խ SOLUTION Since a ෇ s͑Ϫ2͒2 ϩ 3 2 ϩ 1 2 ෇ s14, the scalar projection of b onto a is compa b ෇ aؒb ͑Ϫ2͒͑1͒ ϩ 3͑1͒ ϩ 1͑2͒ 3 ෇ ෇ a s14 s14 Խ Խ The vector projection is this scalar projection times the unit vector in the direction of a: proja b ෇ F Q D FIGURE 6 Խ Խ Խ Խ W ෇ ( F cos ␪) D S P Խ Խ ʹ One use of projections occurs in physics in calculating work. In Section 6.4 we defined the work done by a constant force F in moving an object through a distance d as W ෇ Fd, but this applies only when the force is directed along the line of motion of the object. l Suppose, however, that the constant force is a vector F ෇ PR pointing in some other direction as in Figure 6. If the force moves the object from P to Q, then the displacement l vector is D ෇ PQ. The work done by this force is defined to be the product of the component of the force along D and the distance moved: R ¨ ͳ 3 a 3 3 9 3 ෇ a෇ Ϫ , , 14 7 14 14 s14 a But then, from Theorem 3, we have 12 Խ ԽԽ D Խ cos ␪ ෇ F ؒ D W෇ F 5E-13(pp 848-857) ❙❙❙❙ 848 1/18/06 11:18 AM Page 848 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE Thus, the work done by a constant force F is the dot product F ؒ D, where D is the displacement vector. EXAMPLE 7 A crate is hauled 8 m up a ramp under a constant force of 200 N applied at an angle of 25Њ to the ramp. Find the work done. F 25° SOLUTION If F and D are the force and displacement vectors, as pictured in Figure 7, then the work done is D Խ ԽԽ D Խ cos 25Њ W෇FؒD෇ F ෇ ͑200͒͑8͒ cos 25Њ Ϸ 1450 Nиm ෇ 1450 J FIGURE 7 EXAMPLE 8 A force is given by a vector F ෇ 3i ϩ 4j ϩ 5k and moves a particle from the point P͑2, 1, 0͒ to the point Q͑4, 6, 2͒. Find the work done. l SOLUTION The displacement vector is D ෇ PQ ෇ ͗2, 5, 2 ͘ , so by Equation 12, the work done is W ෇ F ؒ D ෇ ͗3, 4, 5͘ ؒ ͗2, 5, 2 ͘ ෇ 6 ϩ 20 ϩ 10 ෇ 36 If the unit of length is meters and the magnitude of the force is measured in newtons, then the work done is 36 joules. |||| 13.3 Exercises 1. Which of the following expressions are meaningful? Which are meaningless? Explain. (a) ͑a ؒ b͒ ؒ c (c) a ͑b ؒ c͒ (e) a ؒ b ϩ c |||| If u is a unit vector, find u ؒ v and u ؒ w. 12. 11. (b) ͑a ؒ b͒c (d) a ؒ ͑b ϩ c͒ (f) a ؒ ͑b ϩ c͒ Խ Խ 11–12 Խ Խ u u v v w 2. Find the dot product of two vectors if their lengths are 6 and and the angle between them is ␲͞4. 1 3 w 3–10 |||| Find a ؒ b. 3. a ෇ ͗4, Ϫ1 ͘ , 4. a ෇ ͗ 1 , 4͘, 2 b ෇ ͗3, 6 ͘ ■ b ෇ ͗Ϫ8, Ϫ3͘ 6. a ෇ ͗s, 2s, 3s͘ , b ෇ ͗t, Ϫt, 5t͘ 10. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 15–20 |||| Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) the angle between a and b is ␲͞6 15. a ෇ ͗3, 4 ͘ , the angle between a and b is 120° ■ ■ ■ drinks on a given day. He charges $2 for a hamburger, $1.50 for a hot dog, and $1 for a soft drink. If A ෇ ͗a, b, c͘ and P ෇ ͗2, 1.5, 1 ͘ , what is the meaning of the dot product A ؒ P ? b ෇ 2i ϩ 4 j ϩ 6k Խ Խ Խ b Խ ෇ 15, Խ a Խ ෇ 4, Խ b Խ ෇ 10, ■ 14. A street vendor sells a hamburgers, b hot dogs, and c soft b ෇ 5i ϩ 9k 9. a ෇ 12, ■ (b) Show that i ؒ i ෇ j ؒ j ෇ k ؒ k ෇ 1. b ෇ ͗3, Ϫ1, 10͘ 8. a ෇ 4 j Ϫ 3 k, ■ 13. (a) Show that i ؒ j ෇ j ؒ k ෇ k ؒ i ෇ 0. 5. a ෇ ͗5, 0, Ϫ2͘ , 7. a ෇ i Ϫ 2 j ϩ 3 k , ■ ■ ■ ■ ■ ■ ■ 16. a ෇ ͗ s3, 1 ͘ , b ෇ ͗5, 12͘ b ෇ ͗0, 5 ͘ 5E-13(pp 848-857) 1/18/06 11:18 AM Page 849 SECTION 13.3 THE DOT PRODUCT 17. a ෇ ͗1, 2, 3͘ , b ෇ ͗4, 0, Ϫ1 ͘ 18. a ෇ ͗6, Ϫ3, 2 ͘ , 19. a ෇ j ϩ k, 37. a ෇ ͗4, 2, 0͘ , 39. a ෇ i ϩ k , b ෇ i ϩ 2 j Ϫ 3k 20. a ෇ 2 i Ϫ j ϩ k, ■ ■ ■ 21–22 ■ ■ ■ ■ ■ ■ ■ ■ ■ Find, correct to the nearest degree, the three angles of the triangle with the given vertices. B͑3, 6͒, 22. D͑0, 1, 1͒, ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 41. Show that the vector orth a b ෇ b Ϫ proj a b is orthogonal to a. (It is called an orthogonal projection of b.) drawing the vectors a, b, proj a b, and orth a b. F͑1, 2, Ϫ1͒ ■ ■ b ෇ i ϩ 6 j Ϫ 2k 42. For the vectors in Exercise 36, find orth a b and illustrate by C͑Ϫ1, 4͒ E͑Ϫ2, 4, 3͒, b෇iϪj ■ |||| 21. A͑1, 0͒, b ෇ ͗3, 3, 4 ͘ 40. a ෇ 2 i Ϫ 3 j ϩ k , b ෇ 3i ϩ 2 j Ϫ k 849 b ෇ ͗1, 1, 1͘ 38. a ෇ ͗Ϫ1, Ϫ2, 2 ͘ , b ෇ ͗2, 1, Ϫ2 ͘ ❙❙❙❙ ■ ■ 43. If a ෇ ͗3, 0, Ϫ1͘ , find a vector b such that comp a b ෇ 2. ■ ■ ■ ■ 44. Suppose that a and b are nonzero vectors. 23–24 (a) Under what circumstances is comp a b ෇ comp b a? (b) Under what circumstances is proj a b ෇ proj b a? Determine whether the given vectors are orthogonal, parallel, or neither. |||| 23. (a) a ෇ ͗Ϫ5, 3, 7 ͘ , b ෇ ͗6, Ϫ8, 2 ͘ (b) a ෇ ͗4, 6 ͘ , b ෇ ͗Ϫ3, 2 ͘ (c) a ෇ Ϫi ϩ 2 j ϩ 5 k, b ෇ 3 i ϩ 4 j Ϫ k (d) a ෇ 2 i ϩ 6 j Ϫ 4 k, b ෇ Ϫ3 i Ϫ 9 j ϩ 6 k 45. A constant force with vector representation F ෇ 10 i ϩ 18 j Ϫ 6 k moves an object along a straight line from the point ͑2, 3, 0͒ to the point ͑4, 9, 15͒. Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons. 24. (a) u ෇ ͗Ϫ3, 9, 6͘ , v ෇ ͗4, Ϫ12, Ϫ8͘ (b) u ෇ i Ϫ j ϩ 2 k, v ෇ 2 i Ϫ j ϩ k (c) u ෇ ͗a, b, c͘ , v ෇ ͗Ϫb, a, 0 ͘ ■ ■ ■ ■ ■ ■ ■ ■ 46. Find the work done by a force of 20 lb acting in the direction N50Њ W in moving an object 4 ft due west. ■ ■ ■ ■ 47. A woman exerts a horizontal force of 25 lb on a crate as she pushes it up a ramp that is 10 ft long and inclined at an angle of 20Њ above the horizontal. Find the work done on the box. 25. Use vectors to decide whether the triangle with vertices P͑1, Ϫ3, Ϫ2͒, Q͑2, 0, Ϫ4͒, and R͑6, Ϫ2, Ϫ5͒ is right-angled. 48. A wagon is pulled a distance of 100 m along a horizontal path 26. For what values of b are the vectors ͗Ϫ6, b, 2 ͘ and ͗b, b , b͘ by a constant force of 50 N. The handle of the wagon is held at an angle of 30Њ above the horizontal. How much work is done? 2 orthogonal? 49. Use a scalar projection to show that the distance from a point 27. Find a unit vector that is orthogonal to both i ϩ j and i ϩ k. P1͑x 1, y1͒ to the line a x ϩ by ϩ c ෇ 0 is 28. Find two unit vectors that make an angle of 60Њ with Խ ax 1 ϩ by1 ϩ c sa 2 ϩ b 2 v ෇ ͗3, 4 ͘ . 29–33 |||| Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) Use this formula to find the distance from the point ͑Ϫ2, 3͒ to the line 3x Ϫ 4y ϩ 5 ෇ 0. 29. ͗3, 4, 5 ͘ 50. If r ෇ ͗x, y, z ͘, a ෇ ͗a 1, a 2 , a 3 ͘ , and b ෇ ͗b1, b2 , b3 ͘ , show 30. ͗1, Ϫ2, Ϫ1 ͘ that the vector equation ͑r Ϫ a͒ ؒ ͑r Ϫ b͒ ෇ 0 represents a sphere, and find its center and radius. 31. 2 i ϩ 3 j Ϫ 6 k 51. Find the angle between a diagonal of a cube and one of its 32. 2 i Ϫ j ϩ 2 k 33. ͗c, c, c͘ , ■ ■ edges. where c Ͼ 0 ■ ■ ■ 52. Find the angle between a diagonal of a cube and a diagonal of ■ ■ ■ ■ ■ ■ ■ 34. If a vector has direction angles ␣ ෇ ␲͞4 and ␤ ෇ ␲͞3, find the third direction angle ␥. 35–40 |||| Find the scalar and vector projections of b onto a. 35. a ෇ ͗3, Ϫ4 ͘ , 36. a ෇ ͗1, 2 ͘ , Խ b ෇ ͗5, 0 ͘ b ෇ ͗Ϫ4, 1 ͘ one of its faces. 53. A molecule of methane, CH 4, is structured with the four hydro- gen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the H— C—H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about 109.5Њ. [Hint: Take the vertices of the tetrahedron to be the points ͑1, 0, 0͒, ͑0, 1, 0͒ , 5E-13(pp 848-857) 850 ❙❙❙❙ 1/18/06 11:19 AM Page 850 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE ͑0, 0, 1͒, and ͑1, 1, 1͒ as shown in the figure. Then the centroid is ( 1 , 1 , 1 ).] 2 2 2 57. Use Theorem 3 to prove the Cauchy-Schwarz Inequality: Խa ؒ bԽ ഛ ԽaԽԽbԽ z H 58. The Triangle Inequality for vectors is C H Խa ϩ bԽ ഛ ԽaԽ ϩ ԽbԽ H (a) Give a geometric interpretation of the Triangle Inequality. (b) Use the Cauchy-Schwarz Inequality from Exercise 57 to prove the Triangle Inequality. [Hint: Use the fact that a ϩ b 2 ෇ ͑a ϩ b͒ и ͑a ϩ b͒ and use Property 3 of the dot product.] y H x Խ Խ Խ Խ Խ 54. If c ෇ a b ϩ b a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b. Խ 59. The Parallelogram Law states that 55. Prove Properties 2, 4, and 5 of the dot product (Theorem 2). Խa ϩ bԽ 2 56. Suppose that all sides of a quadrilateral are equal in length and Խ 2 Խ Խ ෇2 a 2 Խ Խ ϩ2 b 2 (a) Give a geometric interpretation of the Parallelogram Law. (b) Prove the Parallelogram Law. (See the hint in Exercise 58.) opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular. |||| 13.4 Խ ϩ aϪb The Cross Product The cross product a ϫ b of two vectors a and b, unlike the dot product, is a vector. For this reason it is also called the vector product. Note that a ϫ b is defined only when a and b are three-dimensional vectors. 1 Definition If a ෇ ͗a 1, a 2 , a 3 ͘ and b ෇ ͗b1, b2 , b3 ͘ , then the cross product of a and b is the vector a ϫ b ෇ ͗a 2 b3 Ϫ a 3 b2 , a 3 b1 Ϫ a 1 b3 , a 1 b2 Ϫ a 2 b1 ͘ This may seem like a strange way of defining a product. The reason for the particular form of Definition 1 is that the cross product defined in this way has many useful properties, as we will soon see. In particular, we will show that the vector a ϫ b is perpendicular to both a and b. In order to make Definition 1 easier to remember, we use the notation of determinants. A determinant of order 2 is defined by Ϳ Ϳ Ϳ Ϳ a c 2 Ϫ6 For example, b ෇ ad Ϫ bc d 1 ෇ 2͑4͒ Ϫ 1͑Ϫ6͒ ෇ 14 4 A determinant of order 3 can be defined in terms of second-order determinants as follows: 2 Խ Խ a1 b1 c1 a2 b2 c2 Ϳ a3 b2 b3 ෇ a1 c2 c3 Ϳ Ϳ b3 b1 Ϫ a2 c3 c1 Ϳ Ϳ b3 b1 ϩ a3 c3 c1 b2 c2 Ϳ 5E-13(pp 848-857) 1/18/06 11:19 AM Page 851 SECTION 13.4 THE CROSS PRODUCT ❙❙❙❙ 851 Observe that each term on the right side of Equation 2 involves a number a i in the first row of the determinant, and a i is multiplied by the second-order determinant obtained from the left side by deleting the row and column in which a i appears. Notice also the minus sign in the second term. For example, Խ Խ 1 3 Ϫ5 Ϳ Ϳ Ϳ Ϫ1 0 1 ෇1 4 2 2 0 4 1 3 Ϫ2 2 Ϫ5 Ϳ Ϳ 1 3 ϩ ͑Ϫ1͒ 2 Ϫ5 0 4 Ϳ ෇ 1͑0 Ϫ 4͒ Ϫ 2͑6 ϩ 5͒ ϩ ͑Ϫ1͒͑12 Ϫ 0͒ ෇ Ϫ38 If we now rewrite Definition 1 using second-order determinants and the standard basis vectors i , j, and k, we see that the cross product of the vectors a ෇ a 1 i ϩ a 2 j ϩ a 3 k and b ෇ b 1 i ϩ b 2 j ϩ b 3 k is aϫb෇ 3 Ϳ Ϳ Ϳ a2 b2 a3 a1 iϪ b3 b1 Ϳ Ϳ a3 a1 jϩ b3 b1 Ϳ a2 k b2 In view of the similarity between Equations 2 and 3, we often write Խ Խ i j k a ϫ b ෇ a1 a2 a3 b1 b2 b3 4 Although the first row of the symbolic determinant in Equation 4 consists of vectors, if we expand it as if it were an ordinary determinant using the rule in Equation 2, we obtain Equation 3. The symbolic formula in Equation 4 is probably the easiest way of remembering and computing cross products. EXAMPLE 1 If a ෇ ͗1, 3, 4͘ and b ෇ ͗2, 7, Ϫ5͘ , then ԽͿ Ϳ Խ Ϳ i j k aϫb෇ 1 3 4 2 7 Ϫ5 ෇ 3 7 4 1 iϪ Ϫ5 2 Ϳ Ϳ Ϳ 4 1 jϩ Ϫ5 2 3 k 7 ෇ ͑Ϫ15 Ϫ 28͒ i Ϫ ͑Ϫ5 Ϫ 8͒ j ϩ ͑7 Ϫ 6͒ k ෇ Ϫ43i ϩ 13j ϩ k EXAMPLE 2 Show that a ϫ a ෇ 0 for any vector a in V3. SOLUTION If a ෇ ͗a 1, a 2 , a 3 ͘ , then Խ Խ i a ϫ a ෇ a1 a1 j a2 a2 k a3 a3 ෇ ͑a 2 a 3 Ϫ a 3 a 2͒ i Ϫ ͑a 1 a 3 Ϫ a 3 a 1͒ j ϩ ͑a 1 a 2 Ϫ a 2 a 1͒ k ෇ 0i Ϫ 0j ϩ 0k ෇ 0 5E-13(pp 848-857) 852 ❙❙❙❙ 1/18/06 11:20 AM Page 852 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE One of the most important properties of the cross product is given by the following theorem. 5 Theorem The vector a ϫ b is orthogonal to both a and b. Proof In order to show that a ϫ b is orthogonal to a, we compute their dot product as follows: ͑a ϫ b͒ ؒ a ෇ Ϳ a2 b2 Ϳ Ϳ a3 a1 a1 Ϫ b3 b1 Ϳ Ϳ a3 a1 a2 ϩ b3 b1 Ϳ a2 a3 b2 ෇ a 1͑a 2 b3 Ϫ a 3 b2 ͒ Ϫ a 2͑a 1 b3 Ϫ a 3 b1 ͒ ϩ a 3͑a 1 b2 Ϫ a 2 b1 ͒ ෇ a 1 a 2 b3 Ϫ a 1 b2 a 3 Ϫ a 1 a 2 b3 ϩ b1 a 2 a 3 ϩ a 1 b2 a 3 Ϫ b1 a 2 a 3 ෇0 A similar computation shows that ͑a ϫ b͒ ؒ b ෇ 0. Therefore, a ϫ b is orthogonal to both a and b. axb a ¨ b If a and b are represented by directed line segments with the same initial point (as in Figure 1), then Theorem 5 says that the cross product a ϫ b points in a direction perpendicular to the plane through a and b. It turns out that the direction of a ϫ b is given by the right-hand rule: If the fingers of your right hand curl in the direction of a rotation (through an angle less than 180Њ) from a to b, then your thumb points in the direction of a ϫ b. Now that we know the direction of the vector a ϫ b, the remaining thing we need to complete its geometric description is its length a ϫ b . This is given by the following theorem. Խ 6 Theorem If ␪ is the angle between a and b (so 0 ഛ ␪ ഛ ␲), then FIGURE 1 Visual 13.4 shows how a ϫ b changes as b changes. Խ Խ a ϫ b Խ ෇ Խ a ԽԽ b Խ sin ␪ Proof From the definitions of the cross product and length of a vector, we have Խa ϫ bԽ 2 ෇ ͑a 2 b3 Ϫ a 3 b2͒2 ϩ ͑a 3 b1 Ϫ a 1 b3͒2 ϩ ͑a 1 b2 Ϫ a 2 b1͒2 ෇ a 2 b 2 Ϫ 2a 2 a 3 b2 b3 ϩ a 2 b 2 ϩ a 2 b 2 Ϫ 2a 1 a 3 b1 b3 ϩ a 2 b 2 2 3 3 2 3 1 1 3 ϩ a 2 b 2 Ϫ 2a 1 a 2 b1 b2 ϩ a 2 b 2 1 2 2 1 ෇ ͑a 2 ϩ a 2 ϩ a 2 ͒͑b 2 ϩ b 2 ϩ b 2 ͒ Ϫ ͑a 1 b1 ϩ a 2 b2 ϩ a 3 b3 ͒2 1 2 3 1 2 3 Խ Խ Խ b Խ Ϫ ͑a ؒ b͒ ෇ Խ a Խ Խ b Խ Ϫ Խ a Խ Խ b Խ cos ␪ ෇ Խ a Խ Խ b Խ ͑1 Ϫ cos ␪͒ ෇ Խ a Խ Խ b Խ sin ␪ ෇ a 2 2 2 2 2 2 2 2 2 2 2 2 (by Theorem 13.3.3) 2 2 Taking square roots and observing that ssin 2␪ ෇ sin ␪ because sin ␪ ജ 0 when 0 ഛ ␪ ഛ ␲, we have a ϫ b ෇ a b sin ␪ Խ Geometric characterization of a ϫ b Խ Խ ԽԽ Խ Since a vector is completely determined by its magnitude and direction, we can now say that a ϫ b is the vector that is perpendicular to both a and b, whose orientation is deter- 5E-13(pp 848-857) 1/18/06 11:20 AM Page 853 SECTION 13.4 THE CROSS PRODUCT ❙❙❙❙ 853 Խ ԽԽ b Խ sin ␪. In fact, that is exactly how mined by the right-hand rule, and whose length is a physicists define a ϫ b. 7 Corollary Two nonzero vectors a and b are parallel if and only if aϫb෇0 Proof Two nonzero vectors a and b are parallel if and only if ␪ ෇ 0 or ␲. In either case sin ␪ ෇ 0, so a ϫ b ෇ 0 and therefore a ϫ b ෇ 0. Խ b ͉b͉ sin ¨ ¨ FIGURE 2 Խ The geometric interpretation of Theorem 6 can be seen by looking at Figure 2. If a and b are represented by directed line segments with the same initial point, then they determine a parallelogram with base a , altitude b sin ␪, and area Խ Խ Խ Խ Խ Խ(Խ b Խ sin ␪ ) ෇ Խ a ϫ b Խ A෇ a a Thus, we have the following way of interpreting the magnitude of a cross product. The length of the cross product a ϫ b is equal to the area of the parallelogram determined by a and b. EXAMPLE 3 Find a vector perpendicular to the plane that passes through the points P͑1, 4, 6͒, Q͑Ϫ2, 5, Ϫ1͒, and R͑1, Ϫ1, 1͒. l l l l SOLUTION The vector PQ ϫ PR is perpendicular to both PQ and PR and is therefore perpendicular to the plane through P, Q, and R. We know from (13.2.1) that l PQ ෇ ͑Ϫ2 Ϫ 1͒ i ϩ ͑5 Ϫ 4͒ j ϩ ͑Ϫ1 Ϫ 6͒ k ෇ Ϫ3i ϩ j Ϫ 7k l PR ෇ ͑1 Ϫ 1͒ i ϩ ͑Ϫ1 Ϫ 4͒ j ϩ ͑1 Ϫ 6͒ k ෇ Ϫ5 j Ϫ 5k We compute the cross product of these vectors: Խ i j k l l PQ ϫ PR ෇ Ϫ3 1 Ϫ7 0 Ϫ5 Ϫ5 Խ ෇ ͑Ϫ5 Ϫ 35͒ i Ϫ ͑15 Ϫ 0͒ j ϩ ͑15 Ϫ 0͒ k ෇ Ϫ40 i Ϫ 15 j ϩ 15k So the vector ͗Ϫ40, Ϫ15, 15͘ is perpendicular to the given plane. Any nonzero scalar multiple of this vector, such as ͗Ϫ8, Ϫ3, 3͘ , is also perpendicular to the plane. EXAMPLE 4 Find the area of the triangle with vertices P͑1, 4, 6͒, Q͑Ϫ2, 5, Ϫ1͒, and R͑1, Ϫ1, 1͒. l l SOLUTION In Example 3 we computed that PQ ϫ PR ෇ ͗Ϫ40, Ϫ15, 15͘ . The area of the parallelogram with adjacent sides PQ and PR is the length of this cross product: l l Խ PQ ϫ PR Խ ෇ s͑Ϫ40͒ 2 ϩ ͑Ϫ15͒2 ϩ 15 2 ෇ 5s82 The area A of the triangle PQR is half the area of this parallelogram, that is, 5 s82. 2 5E-13(pp 848-857) 854 ❙❙❙❙ 1/18/06 11:21 AM Page 854 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE If we apply Theorems 5 and 6 to the standard basis vectors i , j, and k using ␪ ෇ ␲͞2, we obtain iϫj෇k jϫk෇i kϫi෇j j ϫ i ෇ Ϫk k ϫ j ෇ Ϫi i ϫ k ෇ Ϫj Observe that iϫj jϫi Thus, the cross product is not commutative. Also i ϫ ͑i ϫ j͒ ෇ i ϫ k ෇ Ϫj whereas ͑i ϫ i͒ ϫ j ෇ 0 ϫ j ෇ 0 So the associative law for multiplication does not usually hold; that is, in general, ͑a ϫ b͒ ϫ c a ϫ ͑b ϫ c͒ However, some of the usual laws of algebra do hold for cross products. The following theorem summarizes the properties of vector products. 8 Theorem If a, b, and c are vectors and c is a scalar, then 1. a ϫ b ෇ Ϫb ϫ a 2. (ca) ϫ b ෇ c(a ϫ b) ෇ a ϫ (cb) 3. a ϫ (b ϩ c) ෇ a ϫ b ϩ a ϫ c 4. (a ϩ b) ϫ c ෇ a ϫ c ϩ b ϫ c 5. a ؒ ͑b ϫ c͒ ෇ ͑a ϫ b͒ ؒ c 6. a ϫ ͑b ϫ c͒ ෇ ͑a ؒ c͒b Ϫ ͑a ؒ b͒c These properties can be proved by writing the vectors in terms of their components and using the definition of a cross product. We give the proof of Property 5 and leave the remaining proofs as exercises. Proof of Property 5 If a ෇ ͗a 1, a 2 , a 3 ͘ , b ෇ ͗b1, b2 , b3 ͘ , and c ෇ ͗c1, c2 , c3 ͘ , then 9 a ؒ ͑b ϫ c͒ ෇ a 1͑b2 c3 Ϫ b3 c2͒ ϩ a 2͑b3 c1 Ϫ b1 c3͒ ϩ a 3͑b1 c2 Ϫ b2 c1͒ ෇ a 1 b2 c3 Ϫ a 1 b3 c2 ϩ a 2 b3 c1 Ϫ a 2 b1 c3 ϩ a 3 b1 c2 Ϫ a 3 b2 c1 ෇ ͑a 2 b3 Ϫ a 3 b2 ͒c1 ϩ ͑a 3 b1 Ϫ a 1 b3 ͒c2 ϩ ͑a 1 b2 Ϫ a 2 b1 ͒c3 ෇ ͑a ϫ b͒ ؒ c The product a ؒ ͑b ϫ c͒ that occurs in Property 5 is called the scalar triple product of the vectors a, b, and c. Notice from Equation 9 that we can write the scalar triple product as a determinant: 10 Խ Խ a1 a ؒ ͑b ϫ c͒ ෇ b1 c1 a2 b2 c2 a3 b3 c3 5E-13(pp 848-857) 1/18/06 11:21 AM Page 855 SECTION 13.4 THE CROSS PRODUCT ❙❙❙❙ 855 The geometric significance of the scalar triple product can be seen by considering the parallelepiped determined by the vectors a, b, and c (Figure 3). The area of the base parallelogram is A ෇ b ϫ c . If ␪ is the angle between a and b ϫ c, then the height h of the parallelepiped is h ෇ a cos ␪ . (We must use cos ␪ instead of cos ␪ in case ␪ Ͼ ␲͞2.) Therefore, the volume of the parallelepiped is bxc Խ h ¨ a c Խ Խ ԽԽ Խ Խ b V ෇ Ah ෇ b ϫ c FIGURE 3 Խ Խ ԽԽ a ԽԽ cos ␪ Խ ෇ Խ a ؒ ͑b ϫ c͒ Խ Thus, we have proved the following formula. 11 The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product: Խ V ෇ a ؒ ͑b ϫ c͒ Խ If we use the formula in (11) and discover that the volume of the parallelepiped determined by a, b, and c is 0, then the vectors must lie in the same plane; that is, they are coplanar. EXAMPLE 5 Use the scalar triple product to show that the vectors a ෇ ͗1, 4, Ϫ7͘ , b ෇ ͗2, Ϫ1, 4͘ , and c ෇ ͗0, Ϫ9, 18͘ are coplanar. SOLUTION We use Equation 10 to compute their scalar triple product: ԽͿ Ϳ Խ Ϳ 1 a ؒ ͑b ϫ c͒ ෇ 2 0 ෇1 4 Ϫ1 Ϫ9 Ϫ1 Ϫ9 Ϫ7 4 18 4 2 Ϫ4 18 0 Ϳ Ϳ 4 2 Ϫ7 18 0 Ϫ1 Ϫ9 Ϳ ෇ 1͑18͒ Ϫ 4͑36͒ Ϫ 7͑Ϫ18͒ ෇ 0 Therefore, by (11) the volume of the parallelepiped determined by a, b, and c is 0. This means that a, b, and c are coplanar. ␶ r ¨ F FIGURE 4 The idea of a cross product occurs often in physics. In particular, we consider a force F acting on a rigid body at a point given by a position vector r. (For instance, if we tighten a bolt by applying a force to a wrench as in Figure 4, we produce a turning effect.) The torque ␶ (relative to the origin) is defined to be the cross product of the position and force vectors ␶෇rϫF and measures the tendency of the body to rotate about the origin. The direction of the torque vector indicates the axis of rotation. According to Theorem 6, the magnitude of the torque vector is Խ ␶ Խ ෇ Խ r ϫ F Խ ෇ Խ r ԽԽ F Խ sin ␪ where ␪ is the angle between the position and force vectors. Observe that the only component of F that can cause a rotation is the one perpendicular to r, that is, F sin ␪. The magnitude of the torque is equal to the area of the parallelogram determined by r and F. Խ Խ 5E-13(pp 848-857) 856 ❙❙❙❙ 1/18/06 11:22 AM Page 856 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 6 A bolt is tightened by applying a 40-N force to a 0.25-m wrench as shown in Figure 5. Find the magnitude of the torque about the center of the bolt. 75° 0.25 m SOLUTION The magnitude of the torque vector is Խ ␶ Խ ෇ Խ r ϫ F Խ ෇ Խ r ԽԽ F Խ sin 75Њ ෇ ͑0.25͒͑40͒ sin 75Њ 40 N ෇ 10 sin 75Њ Ϸ 9.66 Nиm ෇ 9.66 J If the bolt is right-threaded, then the torque vector itself is ␶ ෇ Խ ␶ Խ n Ϸ 9.66 n FIGURE 5 where n is a unit vector directed down into the page. |||| 13.4 Exercises |||| Find the cross product a ϫ b and verify that it is orthogonal to both a and b. 1–7 1. a ෇ ͗1, 2, 0͘ , b ෇ ͗0, 3, 1 ͘ 2. a ෇ ͗5, 1, 4͘ , (b) Use the right-hand rule to decide whether the components of a ϫ b are positive, negative, or 0. b ෇ ͗Ϫ1, 0, 2 ͘ 3. a ෇ 2 i ϩ j Ϫ k, z b ෇ j ϩ 2k 4. a ෇ i Ϫ j ϩ k, b b෇iϩjϩk a 5. a ෇ 3 i ϩ 2 j ϩ 4 k, b ෇ i Ϫ 2 j Ϫ 3k 6. a ෇ i ϩ e t j ϩ eϪt k, b ෇ 2 i ϩ e t j Ϫ eϪt k 7. a ෇ ͗t, t 2, t 3 ͘ , ■ ■ ■ x 13. If a ෇ ͗1, 2, 1͘ and b ෇ ͗0, 1, 3͘ , find a ϫ b and b ϫ a. b ෇ ͗1, 2t, 3t 2 ͘ ■ ■ ■ ■ ■ ■ ■ ■ ■ 8. If a ෇ i Ϫ 2 k and b ෇ j ϩ k, find a ϫ b. Sketch a, b, and 17. Show that 0 ϫ a ෇ 0 ෇ a ϫ 0 for any vector a in V3. 18. Show that ͑a ϫ b͒ ؒ b ෇ 0 for all vectors a and b in V3. 19. Prove Property 1 of Theorem 8. Խ |||| Find u ϫ v and determine whether u ϫ v is directed into the page or out of the page. 20. Prove Property 2 of Theorem 8. 21. Prove Property 3 of Theorem 8. 11. 22. Prove Property 4 of Theorem 8. | u|=6 | u|=5 60° ͑a ϫ b͒ ϫ c. 16. Find two unit vectors orthogonal to both i ϩ j ϩ k and 2 i ϩ k. why. If so, state whether it is a vector or a scalar. (a) a ؒ ͑b ϫ c͒ (b) a ϫ ͑b ؒ c͒ (c) a ϫ ͑b ϫ c͒ (d) ͑a ؒ b͒ ϫ c (e) ͑a ؒ b͒ ϫ ͑c ؒ d͒ (f) ͑a ϫ b͒ ؒ ͑c ϫ d͒ 10. that a ϫ ͑b ϫ c͒ ͗0, 4, 4͘ . 9. State whether each expression is meaningful. If not, explain Խ 14. If a ෇ ͗3, 1, 2͘ , b ෇ ͗Ϫ1, 1, 0͘ , and c ෇ ͗0, 0, Ϫ4 ͘ , show 15. Find two unit vectors orthogonal to both ͗1, Ϫ1, 1͘ and a ϫ b as vectors starting at the origin. 10–11 y | v|=10 23. Find the area of the parallelogram with vertices A͑Ϫ2, 1͒, | v|=8 150° B͑0, 4͒, C͑4, 2͒, and D͑2, Ϫ1͒. 24. Find the area of the parallelogram with vertices K͑1, 2, 3͒, ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 12. The figure shows a vector a in the xy-plane and a vector b in Խ Խ Խ Խ the direction of k. Their lengths are a ෇ 3 and b ෇ 2. (a) Find a ϫ b . Խ Խ ■ L͑1, 3, 6͒, M͑3, 8, 6͒, and N͑3, 7, 3͒. 25–28 |||| (a) Find a vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. 25. P͑1, 0, 0͒, Q͑0, 2, 0͒, R͑0, 0, 3͒ 5E-13(pp 848-857) 1/18/06 11:23 AM Page 857 SECTION 13.4 THE CROSS PRODUCT 26. P͑2, 1, 5͒, 27. P͑0, Ϫ2, 0͒, 28. P͑2, 0, Ϫ3͒, ■ ■ ■ Q͑Ϫ1, 3, 4͒, Q͑3, 1, 0͒, ■ ■ R͑5, 3, 1͒ 39. (a) Let P be a point not on the line L that passes through the R͑5, 2, 2͒ ■ ■ ■ ■ ■ ■ ■ 29. a ෇ ͗6, 3, Ϫ1 ͘ , b ෇ ͗0, 1, 2 ͘ , 30. a ෇ i ϩ j Ϫ k, b ෇ i Ϫ j ϩ k, c ෇ Ϫ i ϩ j ϩ k ■ ■ ■ ■ ■ c ෇ ͗4, Ϫ2, 5͘ ■ ■ ■ ■ ■ ■ 31–32 |||| Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 31. P͑2, 0, Ϫ1͒, 32. P͑0, 1, 2͒, ■ ■ ■ R͑3, Ϫ1, 1͒, Q͑4, 1, 0͒, Q͑2, 4, 5͒, ■ ■ ■ ■ S͑2, Ϫ2, 2͒ ■ ■ ■ ■ ■ 33. Use the scalar triple product to verify that the vectors a ෇ 2 i ϩ 3 j ϩ k, b ෇ i Ϫ j, and c ෇ 7 i ϩ 3 j ϩ 2 k are coplanar. 34. Use the scalar triple product to determine whether the points P͑1, 0, 1͒, Q͑2, 4, 6͒, R͑3, Ϫ1, 2͒, and S͑6, 2, 8͒ lie in the same plane. 35. A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P. Խ Խ Խ l l where a ෇ QR and b ෇ QP. (b) Use the formula in part (a) to find the distance from the point P͑1, 1, 1͒ to the line through Q͑0, 6, 8͒ and R͑Ϫ1, 4, 7͒. 40. (a) Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance d from P to the plane is ͑ a ϫ b͒ ؒ c d෇ aϫb Խ S͑6, Ϫ1, 4͒ R͑Ϫ1, 0, 1͒, points Q and R. Show that the distance d from the point P to the line L is aϫb d෇ a Խ 29–30 |||| Find the volume of the parallelepiped determined by the vectors a, b, and c. ■ 857 minimum values of the length of the vector u ϫ v. In what direction does u ϫ v point? R͑3, 0, 6͒ Q͑4, 1, Ϫ2͒, ❙❙❙❙ Խ Խ Խ l l l where a ෇ QR, b ෇ QS, and c ෇ QP. (b) Use the formula in part (a) to find the distance from the point P͑2, 1, 4͒ to the plane through the points Q͑1, 0, 0͒, R͑0, 2, 0͒, and S͑0, 0, 3͒. 41. Prove that ͑a Ϫ b͒ ϫ ͑a ϩ b͒ ෇ 2͑a ϫ b͒. 42. Prove part 6 of Theorem 8, that is, a ϫ ͑b ϫ c͒ ෇ ͑a ؒ c͒b Ϫ ͑a ؒ b͒c 43. Use Exercise 42 to prove that a ϫ ͑b ϫ c͒ ϩ b ϫ ͑c ϫ a͒ ϩ c ϫ ͑a ϫ b͒ ෇ 0 60 N 44. Prove that 70° ͑a ϫ b͒ ؒ ͑c ϫ d͒ ෇ 10° P Ϳ aؒc bؒc aؒd bؒd Ϳ 45. Suppose that a 36. Find the magnitude of the torque about P if a 36-lb force is applied as shown. P 4 ft 0. (a) If a ؒ b ෇ a ؒ c, does it follow that b ෇ c? (b) If a ϫ b ෇ a ϫ c, does it follow that b ෇ c? (c) If a ؒ b ෇ a ؒ c and a ϫ b ෇ a ϫ c, does it follow that b ෇ c? 46. If v1, v2, and v3 are noncoplanar vectors, let 4 ft 30° 36 lb 37. A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction ͗0, 3, Ϫ4 ͘ at the end of the wrench. Find the magnitude of the force needed to supply 100 J of torque to the bolt. 38. Let v ෇ 5 j and let u be a vector with length 3 that starts at the origin and rotates in the xy-plane. Find the maximum and k1 ෇ v2 ϫ v3 v1 ؒ ͑v2 ϫ v3 ͒ k3 ෇ k2 ෇ v3 ϫ v1 v1 ؒ ͑v2 ϫ v3 ͒ v1 ϫ v2 v1 ؒ ͑v2 ϫ v3 ͒ (These vectors occur in the study of crystallography. Vectors of the form n1 v1 ϩ n 2 v2 ϩ n3 v3 , where each ni is an integer, form a lattice for a crystal. Vectors written similarly in terms of k1, k2, and k3 form the reciprocal lattice.) (a) Show that k i is perpendicular to vj if i j. (b) Show that k i ؒ vi ෇ 1 for i ෇ 1, 2, 3. 1 (c) Show that k1 ؒ ͑k2 ϫ k3 ͒ ෇ . v1 ؒ ͑v2 ϫ v3 ͒ 5E-13(pp 858-867) 858 ❙❙❙❙ 1/18/06 11:26 AM Page 858 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE DISCOVERY PROJECT The Geometry of a Tetrahedron A tetrahedron is a solid with four vertices, P, Q, R, and S, and four triangular faces as shown in the figure. P 1. Let v1 , v2 , v3 , and v4 be vectors with lengths equal to the areas of the faces opposite the vertices P, Q, R, and S, respectively, and directions perpendicular to the respective faces and pointing outward. Show that v1 ϩ v2 ϩ v3 ϩ v4 ෇ 0 S R Q 2. The volume V of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face. (a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices P, Q, R, and S. (b) Find the volume of the tetrahedron whose vertices are P͑1, 1, 1͒, Q͑1, 2, 3͒, R͑1, 1, 2͒, and S͑3, Ϫ1, 2͒. 3. Suppose the tetrahedron in the figure has a trirectangular vertex S. (This means that the three angles at S are all right angles.) Let A, B, and C be the areas of the three faces that meet at S, and let D be the area of the opposite face PQR. Using the result of Problem 1, or otherwise, show that D 2 ෇ A2 ϩ B 2 ϩ C 2 (This is a three-dimensional version of the Pythagorean Theorem.) |||| 13.5 Equations of Lines and Planes z P¸(x¸, y¸, z¸) a P(x, y, z) L r¸ O r v x y FIGURE 1 A line in the xy-plane is determined when a point on the line and the direction of the line (its slope or angle of inclination) are given. The equation of the line can then be written using the point-slope form. Likewise, a line L in three-dimensional space is determined when we know a point P0͑x 0 , y0 , z0͒ on L and the direction of L. In three dimensions the direction of a line is conveniently described by a vector, so we let v be a vector parallel to L. Let P͑x, y, z͒ be an arbitrary point on L and let r0 and r be the position vectors of P0 and P (that is, they have representations OP0 and OP If a is the vector with representation P0 P as in Figure 1, A A). A, then the Triangle Law for vector addition gives r ෇ r0 ϩ a. But, since a and v are parallel vectors, there is a scalar t such that a ෇ tv. Thus 1 z t=0 t>0 L t<0 r¸ x FIGURE 2 y r ෇ r0 ϩ tv which is a vector equation of L. Each value of the parameter t gives the position vector r of a point on L. In other words, as t varies, the line is traced out by the tip of the vector r. As Figure 2 indicates, positive values of t correspond to points on L that lie on one side of P0 , whereas negative values of t correspond to points that lie on the other side of P0 . If the vector v that gives the direction of the line L is written in component form as v ෇ ͗a, b, c͘ , then we have tv ෇ ͗ta, tb, tc͘ . We can also write r ෇ ͗x, y, z͘ and r0 ෇ ͗x 0 , y0 , z0 ͘ , so the vector equation (1) becomes ͗x, y, z͘ ෇ ͗x 0 ϩ ta, y0 ϩ tb, z0 ϩ tc͘ 5E-13(pp 858-867) 1/18/06 11:27 AM Page 859 SECTION 13.5 EQUATIONS OF LINES AND PLANES ❙❙❙❙ 859 Two vectors are equal if and only if corresponding components are equal. Therefore, we have the three scalar equations: 2 x ෇ x 0 ϩ at y ෇ y0 ϩ bt z ෇ z0 ϩ ct where t ʦ ‫ .ޒ‬These equations are called parametric equations of the line L through the point P0͑x 0 , y0 , z0͒ and parallel to the vector v ෇ ͗a, b, c͘ . Each value of the parameter t gives a point ͑x, y, z͒ on L. |||| Figure 3 shows the line L in Example 1 and its relation to the given point and to the vector that gives its direction. z (a) Here r0 ෇ ͗5, 1, 3͘ ෇ 5i ϩ j ϩ 3k and v ෇ i ϩ 4 j Ϫ 2k, so the vector equation (1) becomes r¸ v=i+4j-2k x (a) Find a vector equation and parametric equations for the line that passes through the point ͑5, 1, 3͒ and is parallel to the vector i ϩ 4 j Ϫ 2k. (b) Find two other points on the line. SOLUTION L (5, 1, 3) EXAMPLE 1 r ෇ ͑5i ϩ j ϩ 3k͒ ϩ t͑i ϩ 4 j Ϫ 2k͒ y r ෇ ͑5 ϩ t͒ i ϩ ͑1 ϩ 4t͒ j ϩ ͑3 Ϫ 2t͒ k or Parametric equations are FIGURE 3 x෇5ϩt y ෇ 1 ϩ 4t z ෇ 3 Ϫ 2t (b) Choosing the parameter value t ෇ 1 gives x ෇ 6, y ෇ 5, and z ෇ 1, so ͑6, 5, 1͒ is a point on the line. Similarly, t ෇ Ϫ1 gives the point ͑4, Ϫ3, 5͒. The vector equation and parametric equations of a line are not unique. If we change the point or the parameter or choose a different parallel vector, then the equations change. For instance, if, instead of ͑5, 1, 3͒, we choose the point ͑6, 5, 1͒ in Example 1, then the parametric equations of the line become x෇6ϩt y ෇ 5 ϩ 4t z ෇ 1 Ϫ 2t Or, if we stay with the point ͑5, 1, 3͒ but choose the parallel vector 2i ϩ 8j Ϫ 4k, we arrive at the equations x ෇ 5 ϩ 2t y ෇ 1 ϩ 8t z ෇ 3 Ϫ 4t In general, if a vector v ෇ ͗a, b, c͘ is used to describe the direction of a line L, then the numbers a, b, and c are called direction numbers of L. Since any vector parallel to v could also be used, we see that any three numbers proportional to a, b, and c could also be used as a set of direction numbers for L. Another way of describing a line L is to eliminate the parameter t from Equations 2. If none of a, b, or c is 0, we can solve each of these equations for t, equate the results, and obtain 3 y Ϫ y0 z Ϫ z0 x Ϫ x0 ෇ ෇ a b c 5E-13(pp 858-867) ❙❙❙❙ 860 1/18/06 11:27 AM Page 860 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE These equations are called symmetric equations of L . Notice that the numbers a, b, and c that appear in the denominators of Equations 3 are direction numbers of L, that is, components of a vector parallel to L. If one of a, b, or c is 0, we can still eliminate t. For instance, if a ෇ 0, we could write the equations of L as y Ϫ y0 z Ϫ z0 ෇ b c x ෇ x0 This means that L lies in the vertical plane x ෇ x 0. |||| Figure 4 shows the line L in Example 2 and the point P where it intersects the xy-plane. z 1 B x _1 4 y L (a) We are not explicitly given a vector parallel to the line, but observe that the vector v l with representation AB is parallel to the line and v ෇ ͗3 Ϫ 2, Ϫ1 Ϫ 4, 1 Ϫ ͑Ϫ3͒͘ ෇ ͗1, Ϫ5, 4 ͘ A FIGURE 4 (a) Find parametric equations and symmetric equations of the line that passes through the points A͑2, 4, Ϫ3͒ and B͑3, Ϫ1, 1͒. (b) At what point does this line intersect the xy-plane? SOLUTION 2 1 P EXAMPLE 2 Thus, direction numbers are a ෇ 1, b ෇ Ϫ5, and c ෇ 4. Taking the point ͑2, 4, Ϫ3͒ as P0, we see that parametric equations (2) are x෇2ϩt y ෇ 4 Ϫ 5t z ෇ Ϫ3 ϩ 4t and symmetric equations (3) are xϪ2 yϪ4 zϩ3 ෇ ෇ 1 Ϫ5 4 (b) The line intersects the xy-plane when z ෇ 0, so we put z ෇ 0 in the symmetric equations and obtain xϪ2 yϪ4 3 ෇ ෇ 1 Ϫ5 4 This gives x ෇ 11 and y ෇ 1 , so the line intersects the xy-plane at the point ( 11 , 1 , 0). 4 4 4 4 In general, the procedure of Example 2 shows that direction numbers of the line L through the points P0͑x 0 , y0 , z0 ͒ and P1͑x 1, y1, z1͒ are x 1 Ϫ x 0 , y1 Ϫ y0 , and z1 Ϫ z0 and so symmetric equations of L are x Ϫ x0 y Ϫ y0 z Ϫ z0 ෇ ෇ x1 Ϫ x0 y1 Ϫ y0 z1 Ϫ z0 Often, we need a description, not of an entire line, but of just a line segment. How, for instance, could we describe the line segment AB in Example 2? If we put t ෇ 0 in the parametric equations in Example 2(a), we get the point ͑2, 4, Ϫ3͒ and if we put t ෇ 1 we get ͑3, Ϫ1, 1͒. So the line segment AB is described by the parametric equations x෇2ϩt y ෇ 4 Ϫ 5t z ෇ Ϫ3 ϩ 4t 0ഛtഛ1 or by the corresponding vector equation r͑t͒ ෇ ͗2 ϩ t, 4 Ϫ 5t, Ϫ3 ϩ 4t͘ 0ഛtഛ1 5E-13(pp 858-867) 1/18/06 11:28 AM Page 861 SECTION 13.5 EQUATIONS OF LINES AND PLANES ❙❙❙❙ 861 In general, we know from Equation 1 that the vector equation of a line through the (tip of the) vector r 0 in the direction of a vector v is r ෇ r 0 ϩ tv. If the line also passes through (the tip of) r1, then we can take v ෇ r1 Ϫ r 0 and so its vector equation is r ෇ r 0 ϩ t͑r1 Ϫ r 0͒ ෇ ͑1 Ϫ t͒r 0 ϩ tr1 The line segment from r 0 to r1 is given by the parameter interval 0 ഛ t ഛ 1. 4 The line segment from r 0 to r1 is given by the vector equation r͑t͒ ෇ ͑1 Ϫ t͒r 0 ϩ t r1 |||| The lines L 1 and L 2 in Example 3, shown in Figure 5, are skew lines. EXAMPLE 3 Show that the lines L 1 and L 2 with parametric equations x෇1ϩt 5 L™ y ෇ Ϫ2 ϩ 3t z෇4Ϫt x ෇ 2s z L¡ 0ഛtഛ1 y෇3ϩs z ෇ Ϫ3 ϩ 4s are skew lines; that is, they do not intersect and are not parallel (and therefore do not lie in the same plane). SOLUTION The lines are not parallel because the corresponding vectors ͗1, 3, Ϫ1͘ and 5 10 5 x y ͗2, 1, 4͘ are not parallel. (Their components are not proportional.) If L 1 and L 2 had a point of intersection, there would be values of t and s such that 1 ϩ t ෇ 2s _5 Ϫ2 ϩ 3t ෇ 3 ϩ s 4 Ϫ t ෇ Ϫ3 ϩ 4s FIGURE 5 But if we solve the first two equations, we get t ෇ 11 and s ෇ 8 , and these values don’t 5 5 satisfy the third equation. Therefore, there are no values of t and s that satisfy the three equations. Thus, L 1 and L 2 do not intersect. Hence, L 1 and L 2 are skew lines. Planes Although a line in space is determined by a point and a direction, a plane in space is more difficult to describe. A single vector parallel to a plane is not enough to convey the “direction” of the plane, but a vector perpendicular to the plane does completely specify its direction. Thus, a plane in space is determined by a point P0͑x 0 , y0 , z0͒ in the plane and a vector n that is orthogonal to the plane. This orthogonal vector n is called a normal vector. Let P͑x, y, z͒ be an arbitrary point in the plane, and let r0 and r be the position vectors of P0 and P. Then the vector r Ϫ r0 is represented by P0 P (See Figure 6.) The norA. mal vector n is orthogonal to every vector in the given plane. In particular, n is orthogonal to r Ϫ r0 and so we have z n P (x, y, z) r 0 5 r-r¸ n ؒ ͑r Ϫ r0 ͒ ෇ 0 which can be rewritten as r¸ P¸(x¸, y¸, z¸) x 6 n ؒ r ෇ n ؒ r0 y FIGURE 6 Either Equation 5 or Equation 6 is called a vector equation of the plane. 5E-13(pp 858-867) 862 ❙❙❙❙ 1/18/06 11:28 AM Page 862 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE To obtain a scalar equation for the plane, we write n ෇ ͗a, b, c͘ , r ෇ ͗x, y, z͘ , and r0 ෇ ͗x 0 , y0 , z0 ͘ . Then the vector equation (5) becomes ͗a, b, c͘ ؒ ͗x Ϫ x 0 , y Ϫ y0 , z Ϫ z0 ͘ ෇ 0 or 7 a͑x Ϫ x 0 ͒ ϩ b͑y Ϫ y0 ͒ ϩ c͑z Ϫ z0 ͒ ෇ 0 Equation 7 is the scalar equation of the plane through P0͑x 0 , y0 , z0 ͒ with normal vector n ෇ ͗a, b, c͘ . EXAMPLE 4 Find an equation of the plane through the point ͑2, 4, Ϫ1͒ with normal vector n ෇ ͗2, 3, 4͘ . Find the intercepts and sketch the plane. SOLUTION Putting a ෇ 2, b ෇ 3, c ෇ 4, x 0 ෇ 2, y0 ෇ 4, and z0 ෇ Ϫ1 in Equation 7, we see that an equation of the plane is z (0, 0, 3) 2͑x Ϫ 2͒ ϩ 3͑y Ϫ 4͒ ϩ 4͑z ϩ 1͒ ෇ 0 (0, 4, 0) (6, 0, 0) 2x ϩ 3y ϩ 4z ෇ 12 or y x FIGURE 7 To find the x-intercept we set y ෇ z ෇ 0 in this equation and obtain x ෇ 6. Similarly, the y-intercept is 4 and the z-intercept is 3. This enables us to sketch the portion of the plane that lies in the first octant (see Figure 7). By collecting terms in Equation 7 as we did in Example 4, we can rewrite the equation of a plane as ax ϩ by ϩ cz ϩ d ෇ 0 8 where d ෇ Ϫ͑ax 0 ϩ by0 ϩ cz0 ͒. Equation 8 is called a linear equation in x, y, and z. Conversely, it can be shown that if a, b, and c are not all 0, then the linear equation (8) represents a plane with normal vector ͗a, b, c͘ . (See Exercise 73.) |||| Figure 8 shows the portion of the plane in Example 5 that is enclosed by triangle PQR. EXAMPLE 5 Find an equation of the plane that passes through the points P͑1, 3, 2͒, Q͑3, Ϫ1, 6͒, and R͑5, 2, 0͒. l Q(3, -1, 6) a ෇ ͗2, Ϫ4, 4͘ P(1, 3, 2) b ෇ ͗4, Ϫ1, Ϫ2͘ Since both a and b lie in the plane, their cross product a ϫ b is orthogonal to the plane and can be taken as the normal vector. Thus Խ Խ y i j k n ෇ a ϫ b ෇ 2 Ϫ4 4 ෇ 12 i ϩ 20 j ϩ 14 k 4 Ϫ1 Ϫ2 x R(5, 2, 0) FIGURE 8 l SOLUTION The vectors a and b corresponding to PQ and PR are z With the point P͑1, 3, 2͒ and the normal vector n, an equation of the plane is 12͑x Ϫ 1͒ ϩ 20͑y Ϫ 3͒ ϩ 14͑z Ϫ 2͒ ෇ 0 or 6x ϩ 10y ϩ 7z ෇ 50 5E-13(pp 858-867) 1/18/06 11:29 AM Page 863 SECTION 13.5 EQUATIONS OF LINES AND PLANES ❙❙❙❙ 863 EXAMPLE 6 Find the point at which the line with parametric equations x ෇ 2 ϩ 3t, y ෇ Ϫ4t, z ෇ 5 ϩ t intersects the plane 4x ϩ 5y Ϫ 2z ෇ 18. SOLUTION We substitute the expressions for x, y, and z from the parametric equations into the equation of the plane: 4͑2 ϩ 3t͒ ϩ 5͑Ϫ4t͒ Ϫ 2͑5 ϩ t͒ ෇ 18 This simplifies to Ϫ10t ෇ 20, so t ෇ Ϫ2. Therefore, the point of intersection occurs when the parameter value is t ෇ Ϫ2. Then x ෇ 2 ϩ 3͑Ϫ2͒ ෇ Ϫ4, y ෇ Ϫ4͑Ϫ2͒ ෇ 8, z ෇ 5 Ϫ 2 ෇ 3 and so the point of intersection is ͑Ϫ4, 8, 3͒. n™ ¨ n¡ Two planes are parallel if their normal vectors are parallel. For instance, the planes x ϩ 2y Ϫ 3z ෇ 4 and 2x ϩ 4y Ϫ 6z ෇ 3 are parallel because their normal vectors are n1 ෇ ͗1, 2, Ϫ3͘ and n 2 ෇ ͗2, 4, Ϫ6 ͘ and n 2 ෇ 2n1 . If two planes are not parallel, then they intersect in a straight line and the angle between the two planes is defined as the acute angle between their normal vectors (see angle ␪ in Figure 9). ¨ FIGURE 9 EXAMPLE 7 |||| Figure 10 shows the planes in Example 7 and their line of intersection L. x+y+z=1 x-2y+3z=1 (a) Find the angle between the planes x ϩ y ϩ z ෇ 1 and x Ϫ 2y ϩ 3z ෇ 1. (b) Find symmetric equations for the line of intersection L of these two planes. SOLUTION (a) The normal vectors of these planes are 6 4 2 z 0 _2 _4 n1 ෇ ͗1, 1, 1͘ L n 2 ෇ ͗1, Ϫ2, 3͘ and so, if ␪ is the angle between the planes, Corollary 13.3.6 gives cos ␪ ෇ _2 0 y 2 2 0 x n1 ؒ n 2 1͑1͒ ϩ 1͑Ϫ2͒ ϩ 1͑3͒ 2 ෇ ෇ n1 n 2 s1 ϩ 1 ϩ 1 s1 ϩ 4 ϩ 9 s42 Խ ԽԽ Խ ͩ ͪ _2 ␪ ෇ cosϪ1 FIGURE 10 2 s42 Ϸ 72Њ (b) We first need to find a point on L. For instance, we can find the point where the line intersects the xy-plane by setting z ෇ 0 in the equations of both planes. This gives the equations x ϩ y ෇ 1 and x Ϫ 2y ෇ 1, whose solution is x ෇ 1, y ෇ 0. So the point ͑1, 0, 0͒ lies on L. Now we observe that, since L lies in both planes, it is perpendicular to both of the normal vectors. Thus, a vector v parallel to L is given by the cross product |||| Another way to find the line of intersection is to solve the equations of the planes for two of the variables in terms of the third, which can be taken as the parameter. v ෇ n1 ϫ n 2 ෇ Խ Խ i j k 1 1 1 ෇ 5i Ϫ 2 j Ϫ 3 k 1 Ϫ2 3 and so the symmetric equations of L can be written as xϪ1 y z ෇ ෇ 5 Ϫ2 Ϫ3 NOTE Since a linear equation in x, y, and z represents a plane and two nonparallel planes intersect in a line, it follows that two linear equations can represent a line. The points ͑x, y, z͒ that satisfy both a 1 x ϩ b1 y ϩ c1 z ϩ d1 ෇ 0 and a 2 x ϩ b2 y ϩ c2 z ϩ d2 ෇ 0 ■ 5E-13(pp 858-867) 864 ❙❙❙❙ 1/18/06 11:30 AM CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE L y x-1 = _2 5 2 1 z 0 y _1 2 Page 864 lie on both of these planes, and so the pair of linear equations represents the line of intersection of the planes (if they are not parallel). For instance, in Example 7 the line L was given as the line of intersection of the planes x ϩ y ϩ z ෇ 1 and x Ϫ 2y ϩ 3z ෇ 1. The symmetric equations that we found for L could be written as z xϪ1 y ෇ 5 Ϫ2 =3 and y z ෇ Ϫ2 Ϫ3 _2 _1 y 0 1 2 _2 0 _1 x 1 which is again a pair of linear equations. They exhibit L as the line of intersection of the planes ͑x Ϫ 1͒͞5 ෇ y͑͞Ϫ2͒ and y͑͞Ϫ2͒ ෇ z͑͞Ϫ3͒. (See Figure 11.) In general, when we write the equations of a line in the symmetric form FIGURE 11 |||| Figure 11 shows how the line L in Example 7 can also be regarded as the line of intersection of planes derived from its symmetric equations. x Ϫ x0 y Ϫ y0 z Ϫ z0 ෇ ෇ a b c we can regard the line as the line of intersection of the two planes x Ϫ x0 y Ϫ y0 ෇ a b and y Ϫ y0 z Ϫ z0 ෇ b c EXAMPLE 8 Find a formula for the distance D from a point P1͑x 1, y1, z1͒ to the plane ax ϩ by ϩ cz ϩ d ෇ 0. SOLUTION Let P0͑x 0 , y0 , z0 ͒ be any point in the given plane and let b be the vector corresponding to P0 P1 Then A. b ෇ ͗x 1 Ϫ x 0 , y1 Ϫ y0 , z1 Ϫ z0 ͘ From Figure 12 you can see that the distance D from P1 to the plane is equal to the absolute value of the scalar projection of b onto the normal vector n ෇ ͗a, b, c͘ . (See Section 13.3.) Thus P¡ ¨ b D nؒb Խ Խ ԽnԽ Խ Խ D ෇ compn b ෇ n ෇ Խ a͑x Ϫ x0 ͒ ϩ b͑y1 Ϫ y0 ͒ ϩ c͑z1 Ϫ z0 ͒ sa 2 ϩ b 2 ϩ c 2 ෇ P¸ Խ ͑ax ϩ by1 ϩ cz1 ͒ Ϫ ͑ax0 ϩ by0 ϩ cz0 ͒ sa 2 ϩ b 2 ϩ c 2 FIGURE 12 1 1 Խ Խ Since P0 lies in the plane, its coordinates satisfy the equation of the plane and so we have ax 0 ϩ by0 ϩ cz0 ϩ d ෇ 0. Thus, the formula for D can be written as 9 D෇ Խ ax ϩ by1 ϩ cz1 ϩ d sa 2 ϩ b 2 ϩ c 2 1 Խ EXAMPLE 9 Find the distance between the parallel planes 10x ϩ 2y Ϫ 2z ෇ 5 and 5x ϩ y Ϫ z ෇ 1. SOLUTION First we note that the planes are parallel because their normal vectors ͗10, 2, Ϫ2͘ and ͗5, 1, Ϫ1͘ are parallel. To find the distance D between the planes, we choose any point on one plane and calculate its distance to the other plane. In particular, if we put y ෇ z ෇ 0 in the equation of the first plane, we get 10x ෇ 5 and so 5E-13(pp 858-867) 1/18/06 11:30 AM Page 865 ❙❙❙❙ SECTION 13.5 EQUATIONS OF LINES AND PLANES 865 ( 1 , 0, 0) is a point in this plane. By Formula 9, the distance between ( 1 , 0, 0) and the 2 2 plane 5x ϩ y Ϫ z Ϫ 1 ෇ 0 is D෇ Խ 5( ) ϩ 1͑0͒ Ϫ 1͑0͒ Ϫ 1 Խ ෇ 1 2 s5 2 ϩ 12 ϩ ͑Ϫ1͒2 3 2 3s3 ෇ s3 6 So the distance between the planes is s3͞6. EXAMPLE 10 In Example 3 we showed that the lines L1: x ෇ 1 ϩ t y ෇ Ϫ2 ϩ 3t z෇4Ϫt L 2 : x ෇ 2s y෇3ϩs z ෇ Ϫ3 ϩ 4s are skew. Find the distance between them. SOLUTION Since the two lines L 1 and L 2 are skew, they can be viewed as lying on two parallel planes P1 and P2 . The distance between L 1 and L 2 is the same as the distance between P1 and P2 , which can be computed as in Example 9. The common normal vector to both planes must be orthogonal to both v1 ෇ ͗1, 3, Ϫ1͘ (the direction of L 1 ) and v2 ෇ ͗2, 1, 4͘ (the direction of L 2 ). So a normal vector is n ෇ v1 ϫ v2 ෇ Խ Խ i j k 1 3 Ϫ1 ෇ 13i Ϫ 6 j Ϫ 5k 2 1 4 If we put s ෇ 0 in the equations of L 2 , we get the point ͑0, 3, Ϫ3͒ on L 2 and so an equation for P2 is 13͑x Ϫ 0͒ Ϫ 6͑y Ϫ 3͒ Ϫ 5͑z ϩ 3͒ ෇ 0 13x Ϫ 6y Ϫ 5z ϩ 3 ෇ 0 or If we now set t ෇ 0 in the equations for L 1 , we get the point ͑1, Ϫ2, 4͒ on P1 . So the distance between L 1 and L 2 is the same as the distance from ͑1, Ϫ2, 4͒ to 13x Ϫ 6y ϩ 5z ϩ 3 ෇ 0. By Formula 9, this distance is D෇ |||| 13.5 s13 ϩ ͑Ϫ6͒ ϩ ͑Ϫ5͒ 2 2 2 8 Ϸ 0.53 s230 Exercises 1. Determine whether each statement is true or false. (a) (b) (c) (d) (e) (f) (g) (h) (i) ( j) (k) Խ 13͑1͒ Ϫ 6͑Ϫ2͒ Ϫ 5͑4͒ ϩ 3 Խ ෇ Two lines parallel to a third line are parallel. Two lines perpendicular to a third line are parallel. Two planes parallel to a third plane are parallel. Two planes perpendicular to a third plane are parallel. Two lines parallel to a plane are parallel. Two lines perpendicular to a plane are parallel. Two planes parallel to a line are parallel. Two planes perpendicular to a line are parallel. Two planes either intersect or are parallel. Two lines either intersect or are parallel. A plane and a line either intersect or are parallel. 2–5 |||| Find a vector equation and parametric equations for the line. 2. The line through the point ͑1, 0, Ϫ3͒ and parallel to the vector 2 i Ϫ 4 j ϩ 5 k 3. The line through the point ͑Ϫ2, 4, 10͒ and parallel to the vector ͗3, 1, Ϫ8 ͘ 4. The line through the origin and parallel to the line x ෇ 2t, y ෇ 1 Ϫ t, z ෇ 4 ϩ 3t 5. The line through the point (1, 0, 6) and perpendicular to the plane x ϩ 3y ϩ z ෇ 5 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 5E-13(pp 858-867) ❙❙❙❙ 866 6–12 1/18/06 11:32 AM Page 866 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE |||| Find parametric equations and symmetric equations for the line. 23–38 vector ͗Ϫ2, 1, 5͘ 24. The plane through the point ͑4, 0, Ϫ3͒ and with normal 7. The line through the points ͑1, 3, 2͒ and ͑Ϫ4, 3, 0͒ vector j ϩ 2 k 8. The line through the points ͑6, 1, Ϫ3͒ and ͑2, 4, 5͒ 25. The plane through the point ͑1, Ϫ1, 1͒ and with normal vector iϩjϪk 9. The line through the points (0, 2 , 1) and ͑2, 1, Ϫ3͒ 1 26. The plane through the point ͑Ϫ2, 8, 10͒ and perpendicular to 10. The line through ͑2, 1, 0͒ and perpendicular to both i ϩ j the line x ෇ 1 ϩ t, y ෇ 2t, z ෇ 4 Ϫ 3t and j ϩ k 27. The plane through the origin and parallel to the plane 11. The line through ͑1, Ϫ1, 1͒ and parallel to the line 2x Ϫ y ϩ 3z ෇ 1 x ϩ 2 ෇ 2y ෇ z Ϫ 3 1 28. The plane through the point ͑Ϫ1, 6, Ϫ5͒ and parallel to the 12. The line of intersection of the planes x ϩ y ϩ z ෇ 1 plane x ϩ y ϩ z ϩ 2 ෇ 0 and x ϩ z ෇ 0 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 13. Is the line through ͑Ϫ4, Ϫ6, 1͒ and ͑Ϫ2, 0 Ϫ3͒ parallel to the 29. The plane through the point ͑4, Ϫ2, 3͒ and parallel to the plane 3x Ϫ 7z ෇ 12 30. The plane that contains the line x ෇ 3 ϩ 2t, y ෇ t, z ෇ 8 Ϫ t line through ͑10, 18, 4͒ and ͑5, 3, 14͒ ? and is parallel to the plane 2x ϩ 4y ϩ 8z ෇ 17 31. The plane through the points ͑0, 1, 1͒, ͑1, 0, 1͒, and ͑1, 1, 0͒ 14. Is the line through ͑4, 1, Ϫ1͒ and ͑2, 5, 3͒ perpendicular to the line through ͑Ϫ3, 2, 0͒ and ͑5, 1, 4͒ ? 32. The plane through the origin and the points ͑2, Ϫ4, 6͒ and ͑5, 1, 3͒ 15. (a) Find symmetric equations for the line that passes through 33. The plane through the points ͑3, Ϫ1, 2͒, ͑8, 2, 4͒, and the point ͑0, 2, Ϫ1͒ and is parallel to the line with parametric equations x ෇ 1 ϩ 2t, y ෇ 3t, z ෇ 5 Ϫ 7t. (b) Find the points in which the required line in part (a) intersects the coordinate planes. ͑Ϫ1, Ϫ2, Ϫ3͒ 34. The plane that passes through the point ͑1, 2, 3͒ and contains the line x ෇ 3t, y ෇ 1 ϩ t, z ෇ 2 Ϫ t 16. (a) Find parametric equations for the line through ͑5, 1, 0͒ that is perpendicular to the plane 2x Ϫ y ϩ z ෇ 1. (b) In what points does this line intersect the coordinate planes? 35. The plane that passes through the point ͑6, 0, Ϫ2͒ and contains the line x ෇ 4 Ϫ 2t, y ෇ 3 ϩ 5t, z ෇ 7 ϩ 4 t 36. The plane that passes through the point ͑1, Ϫ1, 1͒ and contains the line with symmetric equations x ෇ 2y ෇ 3z 17. Find a vector equation for the line segment from ͑2, Ϫ1, 4͒ 37. The plane that passes through the point ͑Ϫ1, 2, 1͒ and contains to ͑4, 6, 1͒. the line of intersection of the planes x ϩ y Ϫ z ෇ 2 and 2x Ϫ y ϩ 3z ෇ 1 18. Find parametric equations for the line segment from ͑10, 3, 1͒ to ͑5, 6, Ϫ3͒. 19–22 38. The plane that passes through the line of intersection of the planes x Ϫ z ෇ 1 and y ϩ 2z ෇ 3 and is perpendicular to the plane x ϩ y Ϫ 2z ෇ 1 Determine whether the lines L 1 and L 2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. |||| ■ 19. L 1: x ෇ Ϫ6t, y ෇ 1 ϩ 9t, L 2: x ෇ 1 ϩ 2s, 20. L 1: x ෇ 1 ϩ 2t, L 2: x ෇ Ϫ1 ϩ s, 21. L 1: z ෇ Ϫ3t y ෇ 4 Ϫ 3s, y ෇ 3t, x yϪ1 zϪ2 ෇ ෇ , 1 2 3 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Find the point at which the line intersects the given plane. 41. x ෇ y Ϫ 1 ෇ 2z ; xϪ3 yϪ2 zϪ1 ෇ ෇ Ϫ4 Ϫ3 2 x Ϫ y ϩ 2z ෇ 9 ■ ■ x ϩ 2y Ϫ z ϩ 1 ෇ 0 4x Ϫ y ϩ 3z ෇ 8 ■ ■ ■ ■ ■ ■ ■ ■ 42. Where does the line through ͑1, 0, 1͒ and ͑4, Ϫ2, 2͒ intersect the plane x ϩ y ϩ z ෇ 6 ? 43. Find direction numbers for the line of intersection of the planes x ϩ y ϩ z ෇ 1 and x ϩ z ෇ 0. xϪ2 yϪ6 zϩ2 L 2: ෇ ෇ 1 Ϫ1 3 ■ |||| ■ 40. x ෇ 1 ϩ 2t, y ෇ 4t, z ෇ 2 Ϫ 3t ; z ෇ 1 ϩ 3s L 2: ■ 39. x ෇ 3 Ϫ t, y ෇ 2 ϩ t, z ෇ 5t ; z෇2Ϫt y ෇ 4 ϩ s, ■ 39–41 z෇s xϪ1 yϪ3 zϪ2 ෇ ෇ 22. L 1: 2 2 Ϫ1 ■ Find an equation of the plane. 23. The plane through the point ͑6, 3, 2͒ and perpendicular to the 6. The line through the origin and the point ͑1, 2, 3͒ ■ |||| 44. Find the cosine of the angle between the planes x ϩ y ϩ z ෇ 0 ■ ■ ■ ■ ■ ■ and x ϩ 2y ϩ 3z ෇ 1. 5E-13(pp 858-867) 1/18/06 11:33 AM Page 867 ❙❙❙❙ SECTION 13.5 EQUATIONS OF LINES AND PLANES 45–50 |||| Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. 45. x ϩ 4y Ϫ 3z ෇ 1, 46. 2z ෇ 4y Ϫ x, 49. x ෇ 4y Ϫ 2z, 51–52 ■ L 3 : x ෇ 1 ϩ t, ■ ■ ■ ■ ■ ■ ■ ■ 2x ϩ y ϩ z ෇ 1 ■ ■ ■ ■ ■ ■ ■ 65–66 3x Ϫ 4y ϩ 5z ෇ 6 ■ 63. ͑1, 2, 3͒; ■ ■ ■ ■ ■ 53–54 |||| Find parametric equations for the line of intersection of the planes. ■ ■ ■ ■ ■ ■ ■ |||| ■ ■ ■ ■ ■ ■ 55. Find an equation for the plane consisting of all points that are ■ ■ ■ ■ and z-intercept c. 58. (a) Find the point at which the given lines intersect: r ෇ ͗1, 1, 0 ͘ ϩ t ͗1, Ϫ1, 2 ͘ and r ෇ ͗2, 0, 2 ͘ ϩ s͗Ϫ1, 1, 0͘ (b) Find an equation of the plane that contains these lines. 59. Find parametric equations for the line through the point ͑0, 1, 2͒ that is parallel to the plane x ϩ y ϩ z ෇ 2 and perpendicular to the line x ෇ 1 ϩ t, y ෇ 1 Ϫ t, z ෇ 2t. 60. Find parametric equations for the line through the point ͑0, 1, 2͒ that is perpendicular to the line x ෇ 1 ϩ t, y ෇ 1 Ϫ t, z ෇ 2t and intersects this line. 61. Which of the following four planes are parallel? Are any of them identical? P1: 4 x Ϫ 2y ϩ 6z ෇ 3 P2: 4 x Ϫ 2y Ϫ 2z ෇ 6 P3: Ϫ6 x ϩ 3y Ϫ 9z ෇ 5 P4: z ෇ 2 x Ϫ y Ϫ 3 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Find the distance between the given parallel planes. 3x ϩ 6y Ϫ 3z ෇ 4 ■ ■ x ϩ 2y Ϫ 3z ෇ 1 ■ ■ ■ ■ ■ ■ ■ 69. Show that the distance between the parallel planes ax ϩ by ϩ cz ϩ d1 ෇ 0 and ax ϩ by ϩ cz ϩ d2 ෇ 0 is D෇ 56. Find an equation for the plane consisting of all points that are 57. Find an equation of the plane with x-intercept a, y-intercept b, ■ 4x Ϫ 6y ϩ z ෇ 5 equidistant from the points ͑1, 1, 0͒ and ͑0, 1, 1͒. equidistant from the points ͑Ϫ4, 2, 1͒ and ͑2, Ϫ4, 3͒. z ෇ 1 ϩ 2t ■ x Ϫ 2y Ϫ 2z ෇ 1 68. 3x ϩ 6y Ϫ 9z ෇ 4, ■ y ෇ 3t, ■ z ෇ 5t Find the distance from the point to the given plane. 66. ͑3, Ϫ2, 7͒, ■ y ෇ 2 Ϫ 3t, x ෇ 5 Ϫ t, 67. z ෇ x ϩ 2y ϩ 1, x Ϫ 3y ϩ z ϩ 2 ෇ 0 ■ |||| 67–68 2 x Ϫ 5y Ϫ z ෇ 1 54. 2 x ϩ 5z ϩ 3 ෇ 0, ■ 65. ͑2, 8, 5͒, ■ 53. z ෇ x ϩ y, x ෇ 2 ϩ t, 64. ͑1, 0, Ϫ1͒; (a) Find symmetric equations for the line of intersection of the planes and (b) find the angle between the planes. 52. x Ϫ 2y ϩ z ෇ 1, z෇1Ϫt |||| Use the formula in Exercise 39 in Section 13.4 to find the distance from the point to the given line. |||| 51. x ϩ y Ϫ z ෇ 2, y ෇ 4 ϩ t, 63–64 2x Ϫ y ϩ 2z ෇ 1 ■ z ෇ 2 Ϫ 5t L 4 : r ෇ ͗2, 1, Ϫ3͘ ϩ t ͗2, 2, Ϫ10 ͘ x ϩ 6y ϩ 4z ෇ 3 8y ෇ 1 ϩ 2 x ϩ 4z ■ y ෇ t, L2: x ϩ 1 ෇ y Ϫ 2 ෇ 1 Ϫ z xϪyϩz෇1 50. x ϩ 2y ϩ 2z ෇ 1, ■ identical? L 1 : x ෇ 1 ϩ t, Ϫ3x ϩ 6y ϩ 7z ෇ 0 48. 2x Ϫ 3y ϩ 4z ෇ 5, ■ 62. Which of the following four lines are parallel? Are any of them 3x Ϫ 12y ϩ 6z ෇ 1 47. x ϩ y ϩ z ෇ 1, 867 Խd Խ Ϫ d2 sa ϩ b 2 ϩ c 2 1 2 70. Find equations of the planes that are parallel to the plane x ϩ 2y Ϫ 2z ෇ 1 and two units away from it. 71. Show that the lines with symmetric equations x ෇ y ෇ z and x ϩ 1 ෇ y͞2 ෇ z͞3 are skew, and find the distance between these lines. 72. Find the distance between the skew lines with parametric equations x ෇ 1 ϩ t, y ෇ 1 ϩ 6t, z ෇ 2t, and x ෇ 1 ϩ 2s, y ෇ 5 ϩ 15s, z ෇ Ϫ2 ϩ 6s. 73. If a, b, and c are not all 0, show that the equation ax ϩ by ϩ cz ϩ d ෇ 0 represents a plane and ͗a, b, c͘ is a normal vector to the plane. Hint: Suppose a 0 and rewrite the equation in the form ͩ ͪ a xϩ d a ϩ b͑ y Ϫ 0͒ ϩ c͑z Ϫ 0͒ ෇ 0 74. Give a geometric description of each family of planes. (a) x ϩ y ϩ z ෇ c (b) x ϩ y ϩ cz ෇ 1 (c) y cos ␪ ϩ z sin ␪ ෇ 1 ■ 5E-13(pp 868-877) 868 ❙❙❙❙ 1/18/06 11:34 AM Page 868 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE LABORATORY PROJECT Putting 3D in Perspective Computer graphics programmers face the same challenge as the great painters of the past: how to represent a three-dimensional scene as a flat image on a two-dimensional plane (a screen or a canvas). To create the illusion of perspective, in which closer objects appear larger than those farther away, three-dimensional objects in the computer’s memory are projected onto a rectangular screen window from a viewpoint where the eye, or camera, is located. The viewing volume––the portion of space that will be visible–is the region contained by the four planes that pass through the viewpoint and edge of the screen window. If objects in the scene extend beyond these four planes, they must be truncated before pixel data are sent to the screen. These planes are therefore called clipping planes. 1. Suppose the screen is represented by a rectangle in the yz-plane with vertices ͑0, Ϯ400, 0͒ and ͑0, Ϯ400, 600͒, and the camera is placed at ͑1000, 0, 0͒. A line L in the scene passes through the points ͑230, Ϫ285, 102͒ and ͑860, 105, 264͒. At what points should L be clipped by the clipping planes? 2. If the clipped line segment is projected on the screen window, identify the resulting line segment. 3. Use parametric equations to plot the edges of the screen window, the clipped line segment, and its projection on the screen window. Then add sight lines connecting the viewpoint to each end of the clipped segments to verify that the projection is correct. 4. A rectangle with vertices ͑621, Ϫ147, 206͒, ͑563, 31, 242͒, ͑657, Ϫ111, 86͒, and ͑599, 67, 122͒ is added to the scene. The line L intersects this rectangle. To make the rectangle appear opaque, a programmer can use hidden line rendering which removes portions of objects that are behind other objects. Identify the portion of L that should be removed. |||| 13.6 Cylinders and Quadric Surfaces We have already looked at two special types of surfaces—planes (in Section 13.5) and spheres (in Section 13.1). Here we investigate two other types of surfaces—cylinders and quadric surfaces. In order to sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces (or cross-sections) of the surface. Cylinders A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve. EXAMPLE 1 Sketch the graph of the surface z ෇ x 2. SOLUTION Notice that the equation of the graph, z ෇ x 2, doesn’t involve y. This means that any vertical plane with equation y ෇ k (parallel to the xz-plane) intersects the graph in a curve with equation z ෇ x 2. So these vertical traces are parabolas. Figure 1 shows how the graph is formed by taking the parabola z ෇ x 2 in the xz-plane and moving it in the 5E-13(pp 868-877) 1/18/06 11:34 AM Page 869 SECTION 13.6 CYLINDERS AND QUADRIC SURFACES ❙❙❙❙ 869 direction of the y-axis. The graph is a surface, called a parabolic cylinder, made up of infinitely many shifted copies of the same parabola. Here the rulings of the cylinder are parallel to the y-axis. z FIGURE 1 0 The surface z=≈ is a parabolic cylinder. y x We noticed that the variable y is missing from the equation of the cylinder in Example 1. This is typical of a surface whose rulings are parallel to one of the coordinate axes. If one of the variables x, y, or z is missing from the equation of a surface, then the surface is a cylinder. EXAMPLE 2 Identify and sketch the surfaces. (a) x 2 ϩ y 2 ෇ 1 (b) y 2 ϩ z 2 ෇ 1 SOLUTION (a) Since z is missing and the equations x 2 ϩ y 2 ෇ 1, z ෇ k represent a circle with radius 1 in the plane z ෇ k, the surface x 2 ϩ y 2 ෇ 1 is a circular cylinder whose axis is the z-axis (see Figure 2). Here the rulings are vertical lines. (b) In this case x is missing and the surface is a circular cylinder whose axis is the x-axis (see Figure 3). It is obtained by taking the circle y 2 ϩ z 2 ෇ 1, x ෇ 0 in the yz-plane and moving it parallel to the x-axis. z z y 0 x y x FIGURE 2 ≈+¥=1 | FIGURE 3 ¥+z@=1 NOTE When you are dealing with surfaces, it is important to recognize that an equation like x 2 ϩ y 2 ෇ 1 represents a cylinder and not a circle. The trace of the cylinder x 2 ϩ y 2 ෇ 1 in the xy-plane is the circle with equations x 2 ϩ y 2 ෇ 1, z ෇ 0. ■ Quadric Surfaces A quadric surface is the graph of a second-degree equation in three variables x, y, and z. The most general such equation is Ax 2 ϩ By 2 ϩ Cz 2 ϩ Dxy ϩ Eyz ϩ Fxz ϩ Gx ϩ Hy ϩ Iz ϩ J ෇ 0 5E-13(pp 868-877) 870 ❙❙❙❙ 1/18/06 11:35 AM Page 870 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE where A, B, C, . . . , J are constants, but by translation and rotation it can be brought into one of the two standard forms Ax 2 ϩ By 2 ϩ Cz 2 ϩ J ෇ 0 or Ax 2 ϩ By 2 ϩ Iz ෇ 0 Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane. (See Section 11.5 for a review of conic sections.) EXAMPLE 3 Use traces to sketch the quadric surface with equation x2 ϩ y2 z2 ϩ ෇1 9 4 SOLUTION By substituting z ෇ 0, we find that the trace in the xy-plane is x 2 ϩ y 2͞9 ෇ 1, which we recognize as an equation of an ellipse. In general, the horizontal trace in the plane z ෇ k is x2 ϩ y2 k2 ෇1Ϫ 9 4 z෇k which is an ellipse, provided that k 2 Ͻ 4, that is, Ϫ2 Ͻ k Ͻ 2. Similarly, the vertical traces are also ellipses: z (0, 0, 2) y2 z2 ϩ ෇ 1 Ϫ k2 9 4 0 (1, 0, 0) (0, 3, 0) y x෇k ͑if Ϫ1 Ͻ k Ͻ 1͒ z2 k2 ෇1Ϫ 4 9 y෇k ͑if Ϫ3 Ͻ k Ͻ 3͒ x2 ϩ x FIGURE 4 The ellipsoid ≈+ y@ z@ + =1 9 4 Figure 4 shows how drawing some traces indicates the shape of the surface. It’s called an ellipsoid because all of its traces are ellipses. Notice that it is symmetric with respect to each coordinate plane; this is a reflection of the fact that its equation involves only even powers of x, y, and z. EXAMPLE 4 Use traces to sketch the surface z ෇ 4x 2 ϩ y 2. SOLUTION If we put x ෇ 0, we get z ෇ y 2, so the yz-plane intersects the surface in a parabola. If we put x ෇ k (a constant), we get z ෇ y 2 ϩ 4k 2. This means that if we slice the graph with any plane parallel to the yz-plane, we obtain a parabola that opens upward. Similarly, if y ෇ k, the trace is z ෇ 4x 2 ϩ k 2, which is again a parabola that opens upward. If we put z ෇ k, we get the horizontal traces 4x 2 ϩ y 2 ෇ k, which we recognize as a family of ellipses. Knowing the shapes of the traces, we can sketch the graph in Figure 5. Because of the elliptical and parabolic traces, the quadric surface z ෇ 4x 2 ϩ y 2 is called an elliptic paraboloid. z FIGURE 5 The surface z=4≈+¥ is an elliptic paraboloid. Horizontal traces are ellipses; vertical traces are parabolas. 0 x y 5E-13(pp 868-877) 1/18/06 11:35 AM Page 871 ❙❙❙❙ SECTION 13.6 CYLINDERS AND QUADRIC SURFACES 871 EXAMPLE 5 Sketch the surface z ෇ y 2 Ϫ x 2. SOLUTION The traces in the vertical planes x ෇ k are the parabolas z ෇ y 2 Ϫ k 2, which open upward. The traces in y ෇ k are the parabolas z ෇ Ϫx 2 ϩ k 2, which open downward. The horizontal traces are y 2 Ϫ x 2 ෇ k, a family of hyperbolas. We draw the families of traces in Figure 6, and we show how the traces appear when placed in their correct planes in Figure 7. z z y Ϯ2 0 1 _1 Ϯ1 _1 0 y Ϯ1 FIGURE 6 Vertical traces are parabolas; horizontal traces are hyperbolas. All traces are labeled with the value of k. x x 0 Ϯ2 1 Traces in x=k are z=¥-k @ Traces in y=k are z=_≈+k @ Traces in z=k are ¥-≈=k z z z 1 0 y 0 FIGURE 7 1 Traces moved to their correct planes x _1 x _1 Traces in x=k In Module 13.6A you can investigate how traces determine the shape of a surface. y y x _1 0 1 Traces in y=k Traces in z=k In Figure 8 we fit together the traces from Figure 7 to form the surface z ෇ y 2 Ϫ x 2, a hyperbolic paraboloid. Notice that the shape of the surface near the origin resembles that of a saddle. This surface will be investigated further in Section 15.7 when we discuss saddle points. z 0 y x FIGURE 8 The surface z=¥-≈ is a hyperbolic paraboloid. EXAMPLE 6 Sketch the surface z2 x2 ϩ y2 Ϫ ෇ 1. 4 4 SOLUTION The trace in any horizontal plane z ෇ k is the ellipse x2 k2 ϩ y2 ෇ 1 ϩ 4 4 z෇k 5E-13(pp 868-877) 872 ❙❙❙❙ 1/18/06 11:35 AM Page 872 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z but the traces in the xz- and yz-planes are the hyperbolas x2 z2 Ϫ ෇1 4 4 y2 Ϫ and z2 ෇1 4 x෇0 This surface is called a hyperboloid of one sheet and is sketched in Figure 9. (0, 1, 0) (2, 0, 0) y෇0 y x The idea of using traces to draw a surface is employed in three-dimensional graphing software for computers. In most such software, traces in the vertical planes x ෇ k and y ෇ k are drawn for equally spaced values of k, and parts of the graph are eliminated using hidden line removal. Table 1 shows computer-drawn graphs of the six basic types of quadric surfaces in standard form. All surfaces are symmetric with respect to the z-axis. If a quadric surface is symmetric about a different axis, its equation changes accordingly. FIGURE 9 TABLE 1 Graphs of quadric surfaces Surface Equation x2 y2 z2 ෇1 2 ϩ 2 ϩ a b c2 Ellipsoid z Surface Equation x2 y2 z2 2 ෇ 2 ϩ c a b2 Cone z All traces are ellipses. Horizontal traces are ellipses. If a ෇ b ෇ c, the ellipsoid is a sphere. y x x z x2 y2 ෇ 2 ϩ 2 c a b Elliptic Paraboloid z y Vertical traces in the planes x ෇ k and y ෇ k are hyperbolas if k 0 but are pairs of lines if k ෇ 0. y2 z2 x2 ෇1 2 ϩ 2 Ϫ a b c2 Hyperboloid of One Sheet z Horizontal traces are ellipses. Horizontal traces are ellipses. Vertical traces are parabolas. Vertical traces are hyperbolas. The variable raised to the first power indicates the axis of the paraboloid. x x y The axis of symmetry corresponds to the variable whose coefficient is negative. y z x2 y2 ෇ 2 Ϫ 2 c a b Hyperbolic Paraboloid z Hyperboloid of Two Sheets z Ϫ y2 z2 x2 Ϫ 2 ϩ 2 ෇1 a2 b c Horizontal traces are hyperbolas. Vertical traces are parabolas. y x Horizontal traces in z ෇ k are ellipses if k Ͼ c or k Ͻ Ϫc. Vertical traces are hyperbolas. The case where c Ͻ 0 is illustrated. x y The two minus signs indicate two sheets. 5E-13(pp 868-877) 1/18/06 11:36 AM Page 873 SECTION 13.6 CYLINDERS AND QUADRIC SURFACES In Module 13.6B you can see how changing a, b, and c in Table 1 affects the shape of the quadric surface. ❙❙❙❙ 873 EXAMPLE 7 Identify and sketch the surface 4x 2 Ϫ y 2 ϩ 2z 2 ϩ 4 ෇ 0. SOLUTION Dividing by Ϫ4, we first put the equation in standard form: Ϫx 2 ϩ y2 z2 Ϫ ෇1 4 2 Comparing this equation with Table 1, we see that it represents a hyperboloid of two sheets, the only difference being that in this case the axis of the hyperboloid is the y-axis. The traces in the xy- and yz-planes are the hyperbolas Ϫx 2 ϩ z y2 ෇1 4 z෇0 x෇0 The surface has no trace in the xz-plane, but traces in the vertical planes y ෇ k for k Ͼ 2 are the ellipses z2 k2 x2 ϩ ෇ Ϫ1 y෇k 2 4 which can be written as Խ Խ (0, _2, 0) 0 y (0, 2, 0) x y2 z2 Ϫ ෇1 4 2 and x2 2 k Ϫ1 4 FIGURE 10 4≈-¥+2z@+4=0 z2 ͩ ͪ ϩ 2 k Ϫ1 4 2 ෇1 y෇k These traces are used to make the sketch in Figure 10. z EXAMPLE 8 Classify the quadric surface x 2 ϩ 2z 2 Ϫ 6x Ϫ y ϩ 10 ෇ 0. SOLUTION By completing the square we rewrite the equation as 0 y Ϫ 1 ෇ ͑x Ϫ 3͒2 ϩ 2z 2 y Comparing this equation with Table 1, we see that it represents an elliptic paraboloid. Here, however, the axis of the paraboloid is parallel to the y-axis, and it has been shifted so that its vertex is the point ͑3, 1, 0͒. The traces in the plane y ෇ k ͑k Ͼ 1͒ are the ellipses ͑x Ϫ 3͒2 ϩ 2z 2 ෇ k Ϫ 1 y෇k (3, 1, 0) x FIGURE 11 ≈+2z@-6x-y+10=0 |||| 13.6 The trace in the xy-plane is the parabola with equation y ෇ 1 ϩ ͑x Ϫ 3͒2, z ෇ 0. The paraboloid is sketched in Figure 11. Exercises 5. x Ϫ y 2 ෇ 0 1. (a) What does the equation y ෇ x 2 represent as a curve in ‫? 2 ޒ‬ (b) What does it represent as a surface in ‫? 3 ޒ‬ (c) What does the equation z ෇ y 2 represent? 2. (a) Sketch the graph of y ෇ e x as a curve in ‫.2 ޒ‬ (b) Sketch the graph of y ෇ e x as a surface in ‫.3 ޒ‬ (c) Describe and sketch the surface z ෇ e y. 3–8 |||| Describe and sketch the surface. 3. y ϩ 4z 2 ෇ 4 2 4. z ෇ 4 Ϫ x 2 6. yz ෇ 4 7. z ෇ cos x ■ ■ ■ 8. x 2 Ϫ y 2 ෇ 1 ■ ■ ■ ■ ■ ■ ■ ■ ■ 9. (a) Find and identify the traces of the quadric surface x 2 ϩ y 2 Ϫ z 2 ෇ 1 and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. (b) If we change the equation in part (a) to x 2 Ϫ y 2 ϩ z 2 ෇ 1, how is the graph affected? (c) What if we change the equation in part (a) to x 2 ϩ y 2 ϩ 2y Ϫ z 2 ෇ 0? 5E-13(pp 868-877) 874 ❙❙❙❙ 1/18/06 11:37 AM Page 874 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE 10. (a) Find and identify the traces of the quadric surface 29–36 |||| Reduce the equation to one of the standard forms, classify the surface, and sketch it. Ϫx 2 Ϫ y 2 ϩ z 2 ෇ 1 and explain why the graph looks like the graph of the hyperboloid of two sheets in Table 1. (b) If the equation in part (a) is changed to x 2 Ϫ y 2 Ϫ z 2 ෇ 1, what happens to the graph? Sketch the new graph. 29. z 2 ෇ 4x 2 ϩ 9y 2 ϩ 36 31. x ෇ 2y 2 ϩ 3z 2 |||| Find the traces of the given surface in the planes x ෇ k, y ෇ k, z ෇ k. Then identify the surface and sketch it. 11–20 11. 4x 2 ϩ 9y 2 ϩ 36z 2 ෇ 36 16. 25y 2 ϩ z 2 ෇ 100 ϩ 4x 2 17. x 2 ϩ 4z 2 Ϫ y ෇ 0 18. x 2 ϩ 4y 2 ϩ z 2 ෇ 4 19. y ෇ z 2 Ϫ x 2 34. 4y 2 ϩ z 2 Ϫ x Ϫ 16y Ϫ 4z ϩ 20 ෇ 0 14. z ෇ x 2 Ϫ y 2 15. Ϫx 2 ϩ 4y 2 Ϫ z 2 ෇ 4 20. 16x 2 ෇ y 2 ϩ 4z 2 ■ ■ ■ ■ ■ ■ ■ ■ ■ 32. 4x Ϫ y 2 ϩ 4z 2 ෇ 0 33. 4x 2 ϩ y 2 ϩ 4 z 2 Ϫ 4y Ϫ 24z ϩ 36 ෇ 0 12. 4y ෇ x 2 ϩ z 2 13. y 2 ෇ x 2 ϩ z 2 30. x 2 ෇ 2y 2 ϩ 3z 2 35. x 2 Ϫ y 2 ϩ z 2 Ϫ 4x Ϫ 2y Ϫ 2z ϩ 4 ෇ 0 ■ 36. x 2 Ϫ y 2 ϩ z 2 Ϫ 2x ϩ 2y ϩ 4z ϩ 2 ෇ 0 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ; 37–40 ■ ■ 21–28 |||| Match the equation with its graph (labeled I–VIII). Give reasons for your choices. 21. x 2 ϩ 4y 2 ϩ 9z 2 ෇ 1 26. y 2 ෇ x 2 ϩ 2z 2 27. x 2 ϩ 2z 2 ෇ 1 38. 8x 2 ϩ 15y 2 ϩ 5z 2 ෇ 100 39. z 2 ෇ x 2 ϩ 4y 2 24. Ϫx 2 ϩ y 2 Ϫ z 2 ෇ 1 25. y ෇ 2x 2 ϩ z 2 37. z ෇ 3x 2 Ϫ 5y 2 22. 9x 2 ϩ 4y 2 ϩ z 2 ෇ 1 23. x 2 Ϫ y 2 ϩ z 2 ෇ 1 |||| Use a computer with three-dimensional graphing software to graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface. 28. y ෇ x 2 Ϫ z 2 ■ ■ ■ ■ 40. z ෇ y 2 ϩ xy ■ ■ ■ ■ ■ ■ ■ ■ 41. Sketch the region bounded by the surfaces z ෇ sx 2 ϩ y 2 and x 2 ϩ y 2 ෇ 1 for 1 ഛ z ഛ 2. 42. Sketch the region bounded by the paraboloids z ෇ x 2 ϩ y 2 and z ෇ 2 Ϫ x 2 Ϫ y 2. z I z II 43. Find an equation for the surface obtained by rotating the parabola y ෇ x 2 about the y-axis. y x 44. Find an equation for the surface obtained by rotating the line y x x ෇ 3y about the x-axis. 45. Find an equation for the surface consisting of all points that z III are equidistant from the point ͑Ϫ1, 0, 0͒ and the plane x ෇ 1. Identify the surface. z IV 46. Find an equation for the surface consisting of all points P for y x which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface. y x z V y y x z VII z VI x 47. Show that if the point ͑a, b, c͒ lies on the hyperbolic parabo- z VIII loid z ෇ y 2 Ϫ x 2, then the lines with parametric equations x ෇ a ϩ t, y ෇ b ϩ t, z ෇ c ϩ 2͑b Ϫ a͒t and x ෇ a ϩ t, y ෇ b Ϫ t, z ෇ c Ϫ 2͑b ϩ a͒t both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is called a ruled surface; that is, it can be generated by the motion of a straight line. In fact, this exercise shows that through each point on the hyperbolic paraboloid there are two generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperboloids of one sheet.) 48. Show that the curve of intersection of the surfaces x 2 ϩ 2y 2 Ϫ z 2 ϩ 3x ෇ 1 and 2 x 2 ϩ 4y 2 Ϫ 2z 2 Ϫ 5y ෇ 0 lies in a plane. ■ Խ Խ x x ■ 2 2 2 ; 49. Graph the surfaces z ෇ x ϩ y and z ෇ 1 Ϫ y on a common y y ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Խ Խ screen using the domain x ഛ 1.2, y ഛ 1.2 and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the xy-plane is an ellipse. 5E-13(pp 868-877) 1/18/06 11:37 AM Page 875 SECTION 13.7 CYLINDRICAL AND SPHERICAL COORDINATES |||| 13.7 ❙❙❙❙ 875 Cylindrical and Spherical Coordinates Recall that in plane geometry we introduced the polar coordinate system in order to give a convenient description of certain curves and regions. (See Section 11.3.) In three dimensions there are two coordinate systems that are similar to polar coordinates and give convenient descriptions of some commonly occurring surfaces and solids. They will be especially useful in Chapter 16 when we compute volumes and triple integrals. Cylindrical Coordinates z In the cylindrical coordinate system, a point P in three-dimensional space is represented by the ordered triple ͑r, ␪, z͒, where r and ␪ are polar coordinates of the projection of P onto the xy-plane and z is the directed distance from the xy-plane to P (see Figure 1). To convert from cylindrical to rectangular coordinates, we use the equations P (r, ¨, z) z O 1 r ¨ x x ෇ r cos ␪ y ෇ r sin ␪ z෇z y (r, ¨, 0) whereas to convert from rectangular to cylindrical coordinates, we use FIGURE 1 The cylindrical coordinates of a point 2 tan ␪ ෇ r2 ෇ x2 ϩ y2 y x z෇z These equations follow from Equations 11.3.1 and 11.3.2. EXAMPLE 1 (a) Plot the point with cylindrical coordinates ͑2, 2␲͞3, 1͒ and find its rectangular coordinates. (b) Find cylindrical coordinates of the point with rectangular coordinates ͑3, Ϫ3, Ϫ7͒. SOLUTION z (a) The point with cylindrical coordinates ͑2, 2␲͞3, 1͒ is plotted in Figure 2. From Equations 1, its rectangular coordinates are 2π ”2, , 1’ 3 2␲ 1 ෇2 Ϫ 3 2 y ෇ 2 sin 2␲ s3 ෇2 3 2 2 0 2π 3 x FIGURE 2 ͩ ͪ ͩ ͪ x ෇ 2 cos 1 y ෇ Ϫ1 ෇ s3 z෇1 Thus, the point is (Ϫ1, s3, 1) in rectangular coordinates. (b) From Equations 2 we have r ෇ s3 2 ϩ ͑Ϫ3͒2 ෇ 3 s2 tan ␪ ෇ Ϫ3 ෇ Ϫ1 3 z ෇ Ϫ7 so ␪෇ 7␲ ϩ 2n␲ 4 5E-13(pp 868-877) 876 ❙❙❙❙ 1/18/06 11:37 AM Page 876 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z Therefore, one set of cylindrical coordinates is (3s2, 7␲͞4, Ϫ7). Another is (3s2, Ϫ␲͞4, Ϫ7). As with polar coordinates, there are infinitely many choices. 0 (0, c, 0) y (c, 0, 0) x Cylindrical coordinates are useful in problems that involve symmetry about an axis, and the z-axis is chosen to coincide with this axis of symmetry. For instance, the axis of the circular cylinder with Cartesian equation x 2 ϩ y 2 ෇ c 2 is the z-axis. In cylindrical coordinates this cylinder has the very simple equation r ෇ c. (See Figure 3.) This is the reason for the name “cylindrical” coordinates. EXAMPLE 2 Describe the surface whose equation in cylindrical coordinates is z ෇ r. FIGURE 3 SOLUTION The equation says that the z-value, or height, of each point on the surface is the same as r, the distance from the point to the z-axis. Because ␪ doesn’t appear, it can vary. So any horizontal trace in the plane z ෇ k ͑k Ͼ 0͒ is a circle of radius k. These traces suggest that the surface is a cone. This prediction can be confirmed by converting the equation into rectangular coordinates. From the first equation in (2) we have r=c, a cylinder z z2 ෇ r 2 ෇ x 2 ϩ y 2 We recognize the equation z 2 ෇ x 2 ϩ y 2 (by comparison with Table 1 in Section 13.6) as being a circular cone whose axis is the z-axis (see Figure 4). 0 y x EXAMPLE 3 Find an equation in cylindrical coordinates for the ellipsoid FIGURE 4 SOLUTION Since r 2 ෇ x 2 ϩ y 2 from Equations 2, we have 4x 2 ϩ 4y 2 ϩ z 2 ෇ 1. z=r, a cone z 2 ෇ 1 Ϫ 4͑x 2 ϩ y 2 ͒ ෇ 1 Ϫ 4r 2 So an equation of the ellipsoid in cylindrical coordinates is z 2 ෇ 1 Ϫ 4r 2. z P ( ∏, ¨, ˙) Spherical Coordinates ∏ The spherical coordinates ͑ ␳, ␪, ␾͒ of a point P in space are shown in Figure 5, where ␳ ෇ OP is the distance from the origin to P, ␪ is the same angle as in cylindrical coordinates, and ␾ is the angle between the positive z-axis and the line segment OP. Note that ˙ Խ O ¨ Խ y x FIGURE 5 The spherical coordinates of a point ␳ജ0 0ഛ␾ഛ␲ The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. For example, the sphere with center the origin and radius c has the simple equation ␳ ෇ c (see Figure 6); this is the reason for the name “spherical” coordinates. The graph of the equation ␪ ෇ c is a vertical half-plane z z z z c 0 0 0 c y 0 y c y y x x x FIGURE 6 ∏=c, a sphere FIGURE 7 ¨=c, a half-plane FIGURE 8 ˙=c, a half-cone x 0<c<π/2 π/2<c<π 5E-13(pp 868-877) 1/18/06 11:38 AM Page 877 SECTION 13.7 CYLINDRICAL AND SPHERICAL COORDINATES ❙❙❙❙ 877 (see Figure 7), and the equation ␾ ෇ c represents a half-cone with the z-axis as its axis (see Figure 8). The relationship between rectangular and spherical coordinates can be seen from Figure 9. From triangles OPQ and OPPЈ we have z Q z ෇ ␳ cos ␾ P (x, y, z) P (∏, ¨, ˙) z ∏ But x ෇ r cos ␪ and y ෇ r sin ␪, so to convert from spherical to rectangular coordinates, we use the equations ˙ ˙ r ෇ ␳ sin ␾ O x x 3 r ¨ y x ෇ ␳ sin ␾ cos ␪ y ෇ ␳ sin ␾ sin ␪ z ෇ ␳ cos ␾ y P ª(x, y, 0) Also, the distance formula shows that FIGURE 9 ␳2 ෇ x 2 ϩ y 2 ϩ z2 4 We use this equation in converting from rectangular to spherical coordinates. EXAMPLE 4 The point ͑2, ␲͞4, ␲͞3͒ is given in spherical coordinates. Plot the point and find its rectangular coordinates. SOLUTION We plot the point in Figure 10. From Equations 3 we have ␲ ␲ s3 cos ෇2 3 4 2 y ෇ ␳ sin ␾ sin ␪ ෇ 2 sin ␲ ␲ s3 sin ෇2 3 4 2 (2, π/4, π/3) π 3 2 O x π 4 FIGURE 10 ͩ ͪͩ ͪ ͱ ͩ ͪͩ ͪ ͱ x ෇ ␳ sin ␾ cos ␪ ෇ 2 sin z z ෇ ␳ cos ␾ ෇ 2 cos y 1 s2 1 s2 ෇ ෇ 3 2 3 2 ␲ ෇ 2( 1 ) ෇ 1 2 3 Thus, the point ͑2, ␲͞4, ␲͞3͒ is (s3͞2, s3͞2, 1) in rectangular coordinates. EXAMPLE 5 The point (0, 2s3, Ϫ2) is given in rectangular coordinates. Find spherical coordinates for this point. SOLUTION From Equation 4 we have ␳ ෇ sx 2 ϩ y 2 ϩ z 2 ෇ s0 ϩ 12 ϩ 4 ෇ 4 and so Equations 3 give cos ␾ ෇ ␾෇ 2␲ 3 cos ␪ ෇ (Note that ␪ z Ϫ2 1 ෇ ෇Ϫ ␳ 4 2 x ෇0 ␳ sin ␾ ␪෇ ␲ 2 3␲͞2 because y ෇ 2s3 Ͼ 0.) Therefore, spherical coordinates of the given point are ͑4, ␲͞2, 2␲͞3͒. 5E-13(pp 878-883) 878 ❙❙❙❙ 1/18/06 11:45 AM Page 878 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 6 Find an equation in spherical coordinates for the hyperboloid of two sheets with equation x 2 Ϫ y 2 Ϫ z 2 ෇ 1. In Module 13.7 you can investigate families of surfaces in cylindrical and spherical coordinates. SOLUTION Substituting the expressions in Equations 3 into the given equation, we have ␳ 2 sin 2␾ cos 2␪ Ϫ ␳ 2 sin 2␾ sin 2␪ Ϫ ␳ 2 cos 2␾ ෇ 1 ␳ 2 ͓sin 2␾ ͑cos 2␪ Ϫ sin 2␪ ͒ Ϫ cos 2␾͔ ෇ 1 ␳ 2͑sin 2␾ cos 2␪ Ϫ cos 2␾͒ ෇ 1 or EXAMPLE 7 Find a rectangular equation for the surface whose spherical equation is ␳ ෇ sin ␪ sin ␾. SOLUTION From Equations 4 and 3 we have x 2 ϩ y 2 ϩ z 2 ෇ ␳ 2 ෇ ␳ sin ␪ sin ␾ ෇ y x 2 ϩ ( y Ϫ 1 )2 ϩ z 2 ෇ 1 2 4 or which is the equation of a sphere with center (0, 1 , 0) and radius 1 . 2 2 EXAMPLE 8 Use a computer to draw a picture of the solid that remains when a hole of radius 3 is drilled through the center of a sphere of radius 4. |||| Most three-dimensional graphing programs can graph surfaces whose equations are given in cylindrical or spherical coordinates. As Example 8 demonstrates, this is often the most convenient way of drawing a solid. SOLUTION To keep the equations simple, let’s choose the coordinate system so that the center of the sphere is at the origin and the axis of the cylinder that forms the hole is the z-axis. We could use either cylindrical or spherical coordinates to describe the solid, but the description is somewhat simpler if we use cylindrical coordinates. Then the equation of the cylinder is r ෇ 3 and the equation of the sphere is x 2 ϩ y 2 ϩ z 2 ෇ 16, or r 2 ϩ z 2 ෇ 16. The points in the solid lie outside the cylinder and inside the sphere, so they satisfy the inequalities 3 ഛ r ഛ s16 Ϫ z 2 To ensure that the computer graphs only the appropriate parts of these surfaces, we find where they intersect by solving the equations r ෇ 3 and r ෇ s16 Ϫ z 2 : s16 Ϫ z 2 ෇ 3 ? 16 Ϫ z 2 ෇ 9 ? z2 ෇ 7 ? z ෇ Ϯs7 The solid lies between z ෇ Ϫs7 and z ෇ s7, so we ask the computer to graph the surfaces with the following equations and domains: r෇3 Ϫs7 ഛ z ഛ s7 r ෇ s16 Ϫ z 2 0 ഛ ␪ ഛ 2␲ Ϫs7 ഛ z ഛ s7 The resulting picture, shown in Figure 11, is exactly what we want. FIGURE 11 |||| 13.7 0 ഛ ␪ ഛ 2␲ Exercises 1. What are cylindrical coordinates? For what types of surfaces do they provide convenient descriptions? 2. What are spherical coordinates? For what types of surfaces do they provide convenient descriptions? 3–8 |||| Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point. 3. ͑2, ␲͞4, 1͒ 4. ͑1, 3␲͞2, 2͒ 5. ͑3, 0, Ϫ6͒ 6. ͑1, ␲, e͒ 5E-13(pp 878-883) 1/18/06 11:46 AM Page 879 ❙❙❙❙ SECTION 13.7 CYLINDRICAL AND SPHERICAL COORDINATES 7. ͑4, Ϫ␲͞3, 5͒ ■ ■ 9–12 ■ |||| 8. ͑5, ␲͞6, 6͒ ■ ■ ■ ■ ■ ■ ■ ■ 50. x 2 ϩ y 2 ϩ z 2 ෇ 2 10. ͑3, 3, Ϫ2͒ 51. x ෇ 3 52. x 2 ϩ y 2 ϩ z 2 ϩ 2z ෇ 0 12. ͑3, 4, 5͒ 11. (Ϫ1, Ϫs3, 2) ■ ■ 53. x 2 Ϫ y 2 Ϫ 2z 2 ෇ 4 54. y 2 ϩ z 2 ෇ 1 ■ ■ ■ 55. x 2 ϩ y 2 ෇ 2y 56. z ෇ x 2 Ϫ y 2 ■ ■ ■ ■ ■ ■ 13–18 |||| Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. 13. ͑1, 0, 0͒ 15. ͑1, ␲͞6, ␲͞6͒ 18. ͑2, ␲͞4, ␲͞3͒ ■ 19–22 ■ ■ ■ ■ ■ ■ |||| ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0 ഛ ␾ ഛ ␲͞2, ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0 ഛ ␪ ഛ ␲͞2 ␲͞2 ഛ ␾ ഛ ␲ 62. 0 ഛ ␾ ഛ ␲͞3, ■ ■ 0 ഛ ␾ ഛ ␲͞6, 0 ഛ ␳ ഛ sec ␾ ␳ഛ2 ■ ■ ■ ■ ■ ■ ■ ■ 63. A cylindrical shell is 20 cm long, with inner radius 6 cm and outer radius 7 cm. Write inequalities that describe the shell in an appropriate coordinate system. Explain how you have positioned the coordinate system with respect to the shell. 26. ͑4, ␲͞8, 3͒ ■ ■ rഛzഛ2 61. Ϫ␲͞2 ഛ ␪ ഛ ␲͞2, 24. (s6, ␲͞4, s2 ) ■ 59. ␳ ഛ 2, 60. 2 ഛ ␳ ഛ 3, Change from cylindrical to spherical coordinates. 25. (s3, ␲͞2, Ϫ1) ■ Sketch the solid described by the given inequalities. 58. 0 ഛ ␪ ഛ ␲͞2, 22. (Ϫ1, 1, s6 ) ■ 23. (1, ␲͞6, s3 ) ■ ■ |||| ■ 57. r ഛ z ഛ 2 Ϫ r 2 20. (0, s3, 1) 21. ͑0, Ϫ1, Ϫ1͒ 23–26 ■ ■ 2 Change from rectangular to spherical coordinates. |||| 19. (1, s3, 2s3 ) ■ ■ ■ 57–62 16. ͑5, ␲, ␲͞2͒ 17. ͑2, ␲͞3, ␲͞4͒ ■ 14. ͑3, 0, ␲͒ ■ 49–56 |||| Write the equation (a) in cylindrical coordinates and (b) in spherical coordinates. 49. z ෇ x 2 ϩ y 2 Change from rectangular to cylindrical coordinates. 9. ͑1, Ϫ1, 4͒ ■ ■ 879 ■ ■ 64. (a) Find inequalities that describe a hollow ball with diameter 27–30 |||| Change from spherical to cylindrical coordinates. 29. ͑8, ␲͞6, ␲͞2͒ ■ ■ 31–36 ■ |||| 30 cm and thickness 0.5 cm. Explain how you have positioned the coordinate system that you have chosen. (b) Suppose the ball is cut in half. Write inequalities that describe one of the halves. 28. (2 s2, 3␲͞2, ␲͞2) 27. ͑2, 0, 0͒ 30. ͑4, ␲͞4, ␲͞3͒ ■ ■ ■ ■ ■ ■ ■ ■ ■ sphere x 2 ϩ y 2 ϩ z 2 ෇ z. Write a description of the solid in terms of inequalities involving spherical coordinates. Describe in words the surface whose equation is given. 31. r ෇ 3 32. ␳ ෇ 3 33. ␾ ෇ 0 34. ␾ ෇ ␲͞2 35. ␾ ෇ ␲͞3 ■ ■ ■ ; 66. Use a graphing device to draw the solid enclosed by the paraboloids z ෇ x 2 ϩ y 2 and z ෇ 5 Ϫ x 2 Ϫ y 2. 36. ␪ ෇ ␲͞3 ■ ■ ■ ■ ■ ■ 65. A solid lies above the cone z ෇ sx 2 ϩ y 2 and below the ■ ■ ■ ; 67. Use a graphing device to draw a silo consisting of a cylinder with radius 3 and height 10 surmounted by a hemisphere. 37–48 |||| Identify the surface whose equation is given. 37. z ෇ r 2 39. ␳ cos ␾ ෇ 2 40. ␳ sin ␾ ෇ 2 41. r ෇ 2 cos ␪ 42. ␳ ෇ 2 cos ␾ 43. r ϩ z ෇ 25 68. The latitude and longitude of a point P in the Northern Hemi- 38. r ෇ 4 sin ␪ 44. r 2 Ϫ 2z 2 ෇ 4 2 2 45. ␳ 2͑sin 2␾ cos 2␪ ϩ cos 2␾͒ ෇ 4 46. ␳ 2͑sin 2␾ Ϫ 4 cos 2␾͒ ෇ 1 48. ␳ 2 Ϫ 6␳ ϩ 8 ෇ 0 47. r 2 ෇ r ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ sphere are related to spherical coordinates ␳, ␪, ␾ as follows. We take the origin to be the center of the Earth and the positive z -axis to pass through the North Pole. The positive x-axis passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of P is ␣ ෇ 90Њ Ϫ ␾ Њ and the longitude is ␤ ෇ 360Њ Ϫ ␪ Њ. Find the great-circle distance from Los Angeles (lat. 34.06Њ N, long. 118.25Њ W) to Montréal (lat. 45.50Њ N, long. 73.60Њ W). Take the radius of the Earth to be 3960 mi. (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.) 5E-13(pp 878-883) ❙❙❙❙ 880 1/18/06 11:46 AM Page 880 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE LABORATORY PROJECT ; Families of Surfaces In this project you will discover the interesting shapes that members of families of surfaces can take. You will also see how the shape of the surface evolves as you vary the constants. 1. Use a computer to investigate the family of surfaces z ෇ ͑ax 2 ϩ by 2 ͒e Ϫx 2 Ϫy 2 How does the shape of the graph depend on the numbers a and b? 2. Use a computer to investigate the family of surfaces z ෇ x 2 ϩ y 2 ϩ cxy. In particular, you should determine the transitional values of c for which the surface changes from one type of quadric surface to another. 3. Members of the family of surfaces given in spherical coordinates by the equation ␳ ෇ 1 ϩ 0.2 sin m␪ sin n␾ have been suggested as models for tumors and have been called bumpy spheres and wrinkled spheres. Use a computer to investigate this family of surfaces, assuming that m and n are positive integers. What roles do the values of m and n play in the shape of the surface? |||| 13 Review ■ CONCEPT CHECK ■ 1. What is the difference between a vector and a scalar? 12. How do you find the angle between two intersecting planes? 2. How do you add two vectors geometrically? How do you add 13. Write a vector equation, parametric equations, and symmetric them algebraically? 3. If a is a vector and c is a scalar, how is ca related to a geo- metrically? How do you find ca algebraically? 4. How do you find the vector from one point to another? 5. How do you find the dot product a ؒ b of two vectors if you know their lengths and the angle between them? What if you know their components? 6. How are dot products useful? 7. Write expressions for the scalar and vector projections of b onto a. Illustrate with diagrams. 8. How do you find the cross product a ϫ b of two vectors if you know their lengths and the angle between them? What if you know their components? 9. How are cross products useful? 10. (a) How do you find the area of the parallelogram determined by a and b? (b) How do you find the volume of the parallelepiped determined by a, b, and c? 11. How do you find a vector perpendicular to a plane? equations for a line. 14. Write a vector equation and a scalar equation for a plane. 15. (a) How do you tell if two vectors are parallel? (b) How do you tell if two vectors are perpendicular? (c) How do you tell if two planes are parallel? 16. (a) Describe a method for determining whether three points P, Q, and R lie on the same line. (b) Describe a method for determining whether four points P, Q, R, and S lie in the same plane. 17. (a) How do you find the distance from a point to a line? (b) How do you find the distance from a point to a plane? (c) How do you find the distance between two lines? 18. What are the traces of a surface? How do you find them? 19. Write equations in standard form of the six types of quadric surfaces. 20. (a) Write the equations for converting from cylindrical to rectangular coordinates. In what situation would you use cylindrical coordinates? (b) Write the equations for converting from spherical to rectangular coordinates. In what situation would you use spherical coordinates? 5E-13(pp 878-883) 1/18/06 11:46 AM Page 881 CHAPTER 13 REVIEW ■ TRUE-FALSE QUIZ 8. For any vectors u, v, and w in V3, u ϫ ͑v ϫ w͒ ෇ ͑u ϫ v͒ ϫ w. 2. For any vectors u and v in V3, u ϫ v ෇ v ϫ u. 9. For any vectors u and v in V3, ͑u ϫ v͒ ؒ u ෇ 0. Խ 3. For any vectors u and v in V3, u ϫ v ෇ v ϫ u . 10. For any vectors u and v in V3, ͑u ϩ v͒ ϫ v ෇ u ϫ v. 4. For any vectors u and v in V3 and any scalar k, 11. The cross product of two unit vectors is a unit vector. k͑u ؒ v͒ ෇ ͑k u͒ ؒ v. 12. A linear equation Ax ϩ By ϩ Cz ϩ D ෇ 0 represents a line 5. For any vectors u and v in V3 and any scalar k, in space. k͑u ϫ v͒ ෇ ͑k u͒ ϫ v. Խ 13. The set of points {͑x, y, z͒ x 2 ϩ y 2 ෇ 1} is a circle. 6. For any vectors u, v, and w in V3, ͑u ϩ v͒ ϫ w ෇ u ϫ w ϩ v ϫ w. 14. If u ෇ ͗u1, u2 ͘ and v ෇ ͗ v1, v2 ͘ , then u ؒ v ෇ ͗u1v1, u2 v2 ͘ . ■ EXERCISES 1. (a) Find an equation of the sphere with center ͑1, Ϫ1, 2͒ and radius 3. (b) Find the center and radius of the sphere ■ 5. Find the values of x such that the vectors ͗3, 2, x͘ and ͗2x, 4, x͘ are orthogonal. 6. Find two unit vectors that are orthogonal to both j ϩ 2 k x 2 ϩ y 2 ϩ z 2 ϩ 4x ϩ 6y Ϫ 10z ϩ 2 ෇ 0 and i Ϫ 2 j ϩ 3 k. 2. Copy the vectors in the figure and use them to draw each of the following vectors. (a) a ϩ b (c) Ϫ 1 a 2 ■ u ؒ ͑v ϫ w͒ ෇ ͑u ϫ v͒ ؒ w. 1. For any vectors u and v in V3, u ؒ v ෇ v ؒ u. Խ Խ 881 7. For any vectors u, v, and w in V3, Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Խ ❙❙❙❙ 7. Suppose that u ؒ ͑v ϫ w͒ ෇ 2. Find (a) ͑u ϫ v͒ ؒ w (c) v ؒ ͑u ϫ w͒ (b) a Ϫ b (d) 2 a ϩ b (b) u ؒ ͑w ϫ v͒ (d) ͑u ϫ v͒ ؒ v 8. Show that if a, b, and c are in V3, then ͑a ϫ b͒ ؒ ͓͑b ϫ c͒ ϫ ͑c ϫ a͔͒ ෇ ͓a ؒ ͑b ϫ c͔͒ 2 9. Find the acute angle between two diagonals of a cube. a b 10. Given the points A͑1, 0, 1͒, B͑2, 3, 0͒, C͑Ϫ1, 1, 4͒, and 3. If u and v are the vectors shown in the figure, find u ؒ v and Խ u ϫ v Խ. Is u ϫ v directed into the page or out of it? A͑1, 0, 0͒, B͑2, 0, Ϫ1͒, and C͑1, 4, 3͒. (b) Find the area of triangle ABC. the line segment from ͑1, 0, 2͒ to ͑5, 3, 8͒. Find the work done if the distance is measured in meters and the force in newtons. 45° 13. A boat is pulled onto shore using two ropes, as shown in the |u|=2 diagram. If a force of 255 N is needed, find the magnitude of the force in each rope. 4. Calculate the given quantity if (a) (c) (e) (g) (i) (k) 11. (a) Find a vector perpendicular to the plane through the points 12. A constant force F ෇ 3 i 5 j ϩ 10 k moves an object along |v|=3 a ෇ i ϩ j Ϫ 2k D͑0, 3, 2͒, find the volume of the parallelepiped with adjacent edges AB, AC, and AD. b ෇ 3i Ϫ 2 j ϩ k c ෇ j Ϫ 5k Խ Խ 2a ϩ 3b (b) b aؒb (d) a ϫ b (f) a ؒ ͑b ϫ c͒ bϫc (h) a ϫ ͑b ϫ c͒ cϫc ( j) proj a b comp a b The angle between a and b (correct to the nearest degree) Խ Խ 20° 255 N 30° 5E-13(pp 878-883) ❙❙❙❙ 882 1/18/06 11:46 AM Page 882 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE 25. Find the distance between the planes 3x ϩ y Ϫ 4z ෇ 2 14. Find the magnitude of the torque about P if a 50-N force is and 3x ϩ y Ϫ 4z ෇ 24. applied as shown. 26–34 50 N Identify and sketch the graph of each surface. |||| 26. x ෇ 3 27. x ෇ z 28. y ෇ z 30° 29. x 2 ෇ y 2 ϩ 4z 2 2 30. 4x Ϫ y ϩ 2z ෇ 4 40 cm 31. Ϫ4x 2 ϩ y 2 Ϫ 4z 2 ෇ 4 P 15–17 |||| 32. y 2 ϩ z 2 ෇ 1 ϩ x 2 33. 4x 2 ϩ 4y 2 Ϫ 8y ϩ z 2 ෇ 0 Find parametric equations for the line. 15. The line through ͑4, Ϫ1, 2͒ and ͑1, 1, 5͒ 34. x ෇ y 2 ϩ z 2 Ϫ 2y Ϫ 4z ϩ 5 16. The line through ͑1, 0, Ϫ1͒ and parallel to the line 1 3 ■ ͑x Ϫ 4͒ ෇ 1 y ෇ z ϩ 2 2 18–20 ■ |||| ■ ■ ■ ■ ■ ■ ■ Find an equation of the plane. ■ ■ ■ ■ 38. The rectangular coordinates of a point are ͑2, 2, Ϫ1͒. Find the cylindrical and spherical coordinates of the point. ■ 39. The spherical coordinates of a point are ͑8, ␲͞4, ␲͞6͒. Find the ■ ■ ■ ■ ■ ■ ■ 21. Find the point in which the line with parametric equations rectangular and cylindrical coordinates of the point. 40–43 |||| Identify the surface whose equation is given. x ෇ 2 Ϫ t, y ෇ 1 ϩ 3t, z ෇ 4t intersects the plane 2 x Ϫ y ϩ z ෇ 2. 40. ␾ ෇ ␲͞4 22. Find the distance from the origin to the line x ෇ 1 ϩ t, 42. r ෇ cos ␪ y ෇ 2 Ϫ t, z ෇ Ϫ1 ϩ 2t. 23. Determine whether the lines given by the symmetric equations xϪ1 yϪ2 zϪ3 ෇ ෇ 2 3 4 and ■ the rectangular and spherical coordinates of the point. y ෇ 3 Ϫ t, z ෇ 1 ϩ 3t ■ ■ 37. The cylindrical coordinates of a point are (2s3, ␲͞3, 2). Find 20. The plane through ͑1, 2, Ϫ2͒ that contains the line x ෇ 2t, ■ ■ P to the plane y ෇ 1 is twice the distance from P to the point ͑0, Ϫ1, 0͒. Find an equation for this surface and identify it. ■ 19. The plane through ͑3, Ϫ1, 1͒, ͑4, 0, 2͒, and ͑6, 3, 1͒ ■ ■ 36. A surface consists of all points P such that the distance from ■ 18. The plane through ͑2, 1, 0͒ and parallel to x ϩ 4y Ϫ 3z ෇ 1 ■ ■ about the x-axis. Find an equation of the ellipsoid. 2x Ϫ y ϩ 5z ෇ 12 ■ ■ 35. An ellipsoid is created by rotating the ellipse 4x 2 ϩ y 2 ෇ 16 17. The line through ͑Ϫ2, 2, 4͒ and perpendicular to the plane ■ ■ xϩ1 yϪ3 zϩ5 ෇ ෇ 6 Ϫ1 2 are parallel, skew, or intersecting. 24. (a) Show that the planes x ϩ y Ϫ z ෇ 1 and 2x Ϫ 3y ϩ 4z ෇ 5 are neither parallel nor perpendicular. (b) Find, correct to the nearest degree, the angle between these planes. ■ ■ ■ 41. ␪ ෇ ␲͞4 43. ␳ ෇ 3 sec ␾ ■ ■ ■ ■ ■ ■ ■ ■ ■ 44–46 |||| Write the equation in cylindrical coordinates and in spherical coordinates. 44. x 2 ϩ y 2 ෇ 4 45. x 2 ϩ y 2 ϩ z 2 ෇ 4 46. x 2 ϩ y 2 ϩ z 2 ෇ 2 x ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 47. The parabola z ෇ 4y 2, x ෇ 0 is rotated about the z -axis. Write an equation of the resulting surface in cylindrical coordinates. 48. Sketch the solid consisting of all points with spherical coordi- nates ͑ ␳, ␪, ␾͒ such that 0 ഛ ␪ ഛ ␲͞2, 0 ഛ ␾ ഛ ␲͞6, and 0 ഛ ␳ ഛ 2 cos ␾. ■ 5E-13(pp 878-883) 1/18/06 11:46 AM PROBLEMS PLUS Page 883 1. Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r . The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus, the balls are tightly packed in the box. (See the figure.) Find r . (If you have trouble with this problem, read about the problem-solving strategy entitled Use analogy on page 58.) 2. Let B be a solid box with length L, width W, and height H . Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L, W, and H . 3. Let L be the line of intersection of the planes cx ϩ y ϩ z ෇ c and x Ϫ cy ϩ cz ෇ Ϫ1, 1m where c is a real number. (a) Find symmetric equations for L. (b) As the number c varies, the line L sweeps out a surface S. Find an equation for the curve of intersection of S with the horizontal plane z ෇ t (the trace of S in the plane z ෇ t). (c) Find the volume of the solid bounded by S and the planes z ෇ 0 and z ෇ 1. 4. A plane is capable of flying at a speed of 180 km͞h in still air. The pilot takes off from an 1m 1m FIGURE FOR PROBLEM 1 airfield and heads due north according to the plane’s compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actually traveled 80 km at an angle 5° east of north. (a) What is the wind velocity? (b) In what direction should the pilot have headed to reach the intended destination? 5. Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The block’s descent down the plane is slowed by friction; if ␪ is not too large, friction will prevent the block from moving at all. The forces acting on the block are the weight W, where W ෇ mt ( t is the acceleration due to gravity); the normal force N (the normal component of the reactionary force of the plane on the block), where N ෇ n; and the force F due to friction, which acts parallel to the inclined plane, opposing the direction of motion. If the block is at rest and ␪ is increased, F must also increase until ultimately F reaches its maximum, beyond which the block begins to slide. At this angle ␪s , it has been observed that F is proportional to n. Thus, when F is maximal, we can say that F ෇ ␮ s n, where ␮ s is called the coefficient of static friction and depends on the materials that are in contact. (a) Observe that N ϩ F ϩ W ϭ 0 and deduce that ␮ s ෇ tan ␪s . (b) Suppose that, for ␪ Ͼ ␪ s , an additional outside force H is applied to the block, horizontally from the left, and let H ෇ h. If h is small, the block may still slide down the plane; if h is large enough, the block will move up the plane. Let h min be the smallest value of h that allows the block to remain motionless (so that F is maximal). By choosing the coordinate axes so that F lies along the x-axis, resolve each force into components parallel and perpendicular to the inclined plane and show that Խ Խ N F Խ Խ Խ Խ W ¨ FIGURE FOR PROBLEM 5 Խ Խ Խ Խ Խ Խ Խ Խ Խ Խ Խ Խ h min sin ␪ ϩ m t cos ␪ ෇ n (c) Show that and h min cos ␪ ϩ ␮ s n ෇ mt sin ␪ h min ෇ mt tan͑␪ Ϫ ␪s ͒ Does this equation seem reasonable? Does it make sense for ␪ ෇ ␪s ? As ␪ l 90Њ ? Explain. (d) Let h max be the largest value of h that allows the block to remain motionless. (In which direction is F heading?) Show that h max ෇ mt tan͑␪ ϩ ␪s ͒ Does this equation seem reasonable? Explain. 883 ...
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This note was uploaded on 02/08/2010 for the course M 340L taught by Professor Lay during the Spring '10 term at École Normale Supérieure.

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