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Chapter 13 - 5E-13(pp 828-837 11:09 AM Page 828 CHAPTER 13...

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Vectors and the Geometry of Space Wind velocity is a vector because it has both magnitude and direc- tion. Pictured are velocity vectors indicating the wind pattern over San Francisco Bay at 12:00 P . M . on June 11, 2002. C H A P T E R 1 3 5E-13(pp 828-837) 1/18/06 11:09 AM Page 828
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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 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 of real numbers, where is the -coordinate and is the -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 of real numbers. In order to represent points in space, we first choose a fixed point (the origin) and three directed lines through that are perpendicular to each other, called the coordinate axes and labeled the -axis, -axis, and -axis. Usually we think of the - and -axes as being horizontal and the -axis as being vertical, and we draw the orientation of the axes as in Figure 1. The direction of the -axis is determined by the right-hand rule as illus- trated in Figure 2: If you curl the fingers of your right hand around the -axis in the direc- tion of a counterclockwise rotation from the positive -axis to the positive -axis, then your thumb points in the positive direction of the -axis. The three coordinate axes determine the three coordinate planes illustrated in Fig- ure 3(a). The -plane is the plane that contains the - and -axes; the -plane contains the - and -axes; the -plane contains the - and -axes. These three coordinate planes divide space into eight parts, called octants . The first octant , in the foreground, is deter- mined by the positive axes. 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 -plane, the wall on your right is in the -plane, and the floor is in the -plane. The -axis runs along the intersection of the floor and the left wall. The -axis runs along the intersection of the floor and the right wall. The -axis runs up from the floor toward the ceiling along the inter- section 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 z y x xy y z x z FIGURE 3 (a) Coordinate planes y z x O yz -plane xy -plane xz -plane (b) z O right wall left wall y x floor z x x z z y y z y x xy z y x 90 z z z y x z y x O O a , b , c y b x a a , b 829 FIGURE 2 Right-hand rule
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