Chapter 13

# These three coordinate planes divide space into eight

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Unformatted text preview: e contains the x- and z-axes. These three coordinate planes divide space into eight parts, called octants. The ﬁrst octant, in the foreground, is determined by the positive axes. z x z FIGURE 2 Right-hand rule y z-plan ne la xz-p x FIGURE 3 e l al ft w O xy-plane (a) Coordinate planes le y x right w all O floor y (b) 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 xz-plane, the wall on your right is in the yz-plane, and the ﬂoor is in the xy-plane. The x-axis runs along the intersection of the ﬂoor and the left wall. The y-axis runs along the intersection of the ﬂoor and the right wall. The z-axis runs up from the ﬂoor toward the ceiling along the intersection 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. 829 5E-13(pp 828-837) 830 ❙❙❙❙ 1/18/06 11:09 AM Page 830 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z Now if P is any point in space, let a be the (directed) distance from the y z-plane to P, let b be the distance from the xz-plane to P, and let c be the distance from the xy-plane to P. We represent the point P by the ordered triple a, b, c of real numbers and we call a, b, and c the coordinates of P; a is the x-coordinate, b is the y-coordinate, and c is the z-coordinate. Thus, to locate the point a, b, c we can start at the origin O and move a units along the x-axis, then b units parallel to the y-axis, and then c units parallel to the z-axis as in Figure 4. The point P a, b, c determines a rectangular box as in Figure 5. If we drop a perpendicular from P to the xy-plane, we get a point Q with coordinates a, b, 0 called the projection of P on the x y-plane. Similarly, R 0, b, c...
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## This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas.

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