Unformatted text preview: e contains the x and zaxes. 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 Righthand rule
y zplan ne la
xzp x FIGURE 3 e l al
ft w O xyplane
(a) Coordinate planes le
y x right w all O floor y (b) Because many people have some difﬁculty visualizing diagrams of threedimensional
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 xzplane, the
wall on your right is in the yzplane, and the ﬂoor is in the xyplane. The xaxis runs along
the intersection of the ﬂoor and the left wall. The yaxis runs along the intersection of the
ﬂoor and the right wall. The zaxis 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 5E13(pp 828837) 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 zplane to P,
let b be the distance from the xzplane to P, and let c be the distance from the xyplane 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 xcoordinate, b is the ycoordinate, and c is the
zcoordinate. Thus, to locate the point a, b, c we can start at the origin O and move
a units along the xaxis, then b units parallel to the yaxis, and then c units parallel to the
zaxis 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 xyplane, we get a point Q with coordinates a, b, 0 called the projection of P on the x yplane. 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.
 Fall '09
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