Chapter 13

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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. V ectors 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, b 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, c 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 90 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-plan...
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