In this chapter we introduce vectors and coordinate systems forthree-dimensional space. This will be the setting for our study ofthe calculus of functions of two variables in Chapter 15 becausethe graph of such a function is a surface in space. In this chapterwe will see that vectors provide particularly simple descriptionsof lines and planes in space.||||13.1Three-Dimensional Coordinate SystemsTo 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 byan ordered triple of real numbers.In order to represent points in space, we first choose a fixed point (the origin) andthree directed lines through that are perpendicular to each other, called the coordinateaxesand labeled the -axis,-axis, and -axis. Usually we think of the - and -axes asbeing horizontal and the -axis as being vertical, and we draw the orientation of the axesas in Figure 1. The direction of the -axis is determined by the right-hand ruleas 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, thenyour thumb points in the positive direction of the -axis.The three coordinate axes determine the three coordinate planesillustrated in Fig-ure 3(a). The -plane is the plane that contains the - and -axes; the -plane containsthe - and -axes; the -plane contains the - and -axes. These three coordinate planesdivide 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-dimensionalfigures, you may find it helpful to do the following [see Figure 3(b)]. Look at any bottomcorner of a room and call the corner the origin. The wall on your left is in the -plane, thewall on your right is in the -plane, and the floor is in the -plane. The -axis runs alongthe intersection of the floor and the left wall. The -axis runs along the intersection of thefloor 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 sevenother rooms situated in the other seven octants (three on the same floor and four on thefloor below), all connected by the common corner point .OzyxxyyzxzFIGURE 3(a) Coordinate planesyzxOyz-planexy-planexz-plane(b)zOright wallleft wallyxfloorzxxzzyyzyxxyzyx90zzzyxzyxOOa, b, cybxaa, b829FIGURE 2Right-hand rule

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