Chapter 13

# In figure 5 the faces of the rectangular box are

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Unformatted text preview: rallel 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 3 EXAMPLE 2 Describe and sketch the surface in represented by the equation y x. 3 SOLUTION The equation represents the set of all points in whose x- and y-coordinates are equal, that is, x, x, z x . This is a vertical plane that intersects the ,z 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. x 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. Distance Formula in Three Dimensions The distance P1 P2 between the points P1 x 1, y1, z1 and P2 x 2 , y2 , z2 is P1 P2 z P¡(⁄, ›, z¡) x1 2 y2 y1 z2 2 z1 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 A x 2 , y1, z1 and B x 2 , y2 , z1 are the vertices of the box indicated in the ﬁgure, then P™(¤, ﬁ, z™) P1 A x2 x1 AB y2 y1 z2 BP2 z1 Because triangles P1 BP2 and P1 AB are both right-angled, two applications of the Pythagorean Theorem give 0 x s x2 B(¤, ﬁ, z¡) P1 P2 2 P1 B 2 BP2 P1 B A(¤, ›, z¡) 2 P1 A 2 AB 2 y and 2 FIGURE 9 Combining these equations, we get P1 P2 2 P1 A 2 AB 2 x2 P1 P2 x1 2 y2 y1 2 x2 Therefore 2 BP2 x1 2 y2 y1 2 s x2 x1 2 y2 y1 z2 z2 2 z1 z1 z2 2 2 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 P 2, PQ s1 2 2 s1 1, 7 to the point Q 1, 4 3 4 1 2 5 7 3, 5 is 2 3 EXAMPLE 4 Find an equation of a sphere with radius r and center C h, k, l . z SOLUTION By deﬁnition, a sphere is the set of all points P x, y, z whose distance from C is r. (See Figure 10.) Thus, P is on the sphere if and only if PC r. Squaring...
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