Chapter 13

Similarly r 0 b c and s a 0 c are the projections of p

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Unformatted text preview: and S a, 0, c are the projections of P on the y z-plane and xz-plane, 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 one-to-one 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 two-dimensional analytic geometry, the graph of an equation involving x and y is a curve in 2. In three-dimensional 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 z-coordinate is 3. This is the horizontal plane that is parallel to the xy-plane 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 5E-13(pp 828-837) 1/18/06 11:09 AM Page 831 S ECTION 13.1 THREE-DIMENSIONAL COORDINATE SYSTEMS ❙❙❙❙ 831 (b) The equation y 5 represents the set of all points in 3 whose y-coordinate is 5. This is the vertical plane that is parallel to the xz-plane 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 two-dimensional analytic geometry. See Figure 7(b) and (c). In general, if k is a constant, then x k represents a plane parallel to the yz-plane, y k is a plane parallel to the xz-plane, 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.

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