Unformatted text preview: and S a, 0, c are the projections of P on
the y zplane and xzplane, respectively.
As numerical illustrations, the points
4, 3, 5 and 3, 2, 6 are plotted in
Figure 6. P(a, b, c) c O a y x b FIGURE 4 z z z 3 (0, 0, c)
R(0, b, c)
S(a, 0, c) 0 P(a, b, c) _5
x (_4, 3, _5) (0, b, 0) (a, 0, 0) 3 _2 y y x
0 0 _4 _6 y x (3, _2, _6) Q(a, b, 0) FIGURE 5 F IGURE 6 The Cartesian product
x, y, z x, y, z
is the set of all ordered
triples of real numbers and is denoted by 3. We have given a onetoone correspondence between points P in space and ordered triples a, b, c in 3. It is called a threedimensional rectangular coordinate system. Notice that, in terms of coordinates, the
ﬁrst octant can be described as the set of points whose coordinates are all positive.
In twodimensional analytic geometry, the graph of an equation involving x and y is a
curve in 2. In threedimensional analytic geometry, an equation in x, y, and z represents
a surface in 3.
3 EXAMPLE 1 What surfaces in (a) z 3 are represented by the following equations?
(b) y 5 SOLUTION (a) The equation z 3 represents the set x, y, z z 3 , which is the set of all points
in 3 whose zcoordinate is 3. This is the horizontal plane that is parallel to the xyplane
and three units above it as in Figure 7(a).
z z y
5 3
0
x FIGURE 7 0 (a) z=3, a plane in R# y x 5 (b) y=5, a plane in R# 0
y (c) y=5, a line in R@ x 5E13(pp 828837) 1/18/06 11:09 AM Page 831 S ECTION 13.1 THREEDIMENSIONAL COORDINATE SYSTEMS ❙❙❙❙ 831 (b) The equation y 5 represents the set of all points in 3 whose ycoordinate is 5.
This is the vertical plane that is parallel to the xzplane and ﬁve units to the right of it as
in Figure 7(b).
NOTE When an equation is given, we must understand from the context whether it represents a curve in 2 or a surface in 3. In Example 1, y 5 represents a plane in 3, but
of course y 5 can also represent a line in 2 if we are dealing with twodimensional analytic geometry. See Figure 7(b) and (c).
In general, if k is a constant, then x k represents a plane parallel to the yzplane,
y k is a plane parallel to the xzplane, and z k is a plane pa...
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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
 hamrick

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