265L01 - LESSON 1 Vectors in the Plane and in Space READ:...

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LESSON 1 Vectors in the Plane and in Space READ: Sections 13.1, 13.2 NOTES: The notion of a 3-dimensional coordinate system is most likely familiar to you, and you’ve probably met vectors before, perhaps in a physics course. The main purpose of these two sections is to quickly review those ideas, and to establish the notation we’ll be using throughout the rest of the book. The distance between two points in 3-space is given by the length of the diagonal of the rectangular box having those points as opposite corners, as shown in figure 4, page 667 of the text. That diagonal is the hypotenuse of a right triangle. One side of the triangle is an edge of the box, and the other side is the diagonal of one of the faces of the box. The diagonal of the face is itself the hypotenuse of a right triangle. First use the Pythagorean Theorem to compute the length of the face diagonal. Then use the Pythagorean Theorem again to calculate the length of the diagonal of the box. The result is that if P is the point ( x 1 ,y 1 ,z 1 ) and Q is the point ( x 2 ,y 2 ,z 2 ) then the distance between P and Q is given by || PQ || = p ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 . A sphere is made up of all the points at a fixed distance (called the radius of the sphere) from a certain fixed point (that point is called the center of the sphere of course). The distance formula derived above can be used to write down an equation for the sphere with radius r and center at the point C = ( a,b,c ). In fact, the point P = ( x,y,z ) will be on the sphere if and only if || CP || = r , and according to the distance formula above, that is the same as p ( x - a ) 2 + ( y - b ) 2 + ( z - c ) 2 = r , or a little more neatly, ( x - a ) 2 +( y - b ) 2 +( z - c ) 2 = r 2 . That certainly looks a lot like the equation of the 2-space analog of the sphere (that is, the circle). Vectors are used in physics to represent a quantity that has both a magnitude and a direction associated with it (in other words, a quantity that can be meaningfully represented by an arrow). Examples of vector quantities are forces (10 lbs directed downwards, for example) and velocity (3 feet per second, heading northwest). A visual representation of a vector can be provided by an arrow joining two points, say P and Q , in space. The magnitude of the vector is represented by the distance between the points, and the direction of the vector is indicated by the straight line from the first point, P , to the second point, Q . In writing the vector from P to Q , we’ll use the notation --→ PQ , and we’ll draw an arrow from P (the initial point of the vector) to Q (the terminal point of the vector). Two vectors are called equal if they have the same magnitude and the same direction. The particular initial and terminal points are not important as far as the vector is concerned. If two arrows have the same length and are parallel (and pointing in the same
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265L01 - LESSON 1 Vectors in the Plane and in Space READ:...

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