James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 12.1 Three-Dimensional Coordinate Systems 793
S(a, 0, c)
(a, 0, 0)
x
z
0
(0, 0, c)
Q(a, b, 0)
R(0, b, c)
P(a, b, c)
(0, b, 0)
FIGURE 5 FIGURE 6
y
The point Psa, b, cd 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 sa, b, 0d called the projection
of P onto the xy-plane. Similarly, Rs0, b, cd and Ssa, 0, cd are the projections of
P onto the yz-plane and xz-plane, respectively.
As numerical illustrations, the points s24, 3, 25d and s3, 22, 26d are plotted in Figure
6.
x
z
0
_4
3
y
_5
(_4, 3, _5)
x
_6
(3, _2, _6)
The Cartesian product R 3 R 3 R − hsx, y, zd | x, y, z [ Rj is the set of all ordered
triples of real numbers and is denoted by R 3 . We have given a one-to-one correspondence
between points P in space and ordered triples sa, b, cd in R 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.
Surfaces
In two-dimensional analytic geometry, the graph of an equation involving x and y is a
curve in R 2 . In three-dimensional analytic geometry, an equation in x, y, and z represents
a surface in R 3 .
Example 1 What surfaces in R 3 are represented by the following equations?
(a) z − 3 (b) y − 5
SOLUTION
(a) The equation z − 3 represents the set hsx, y, zd | z − 3j, which is the set of all
points in R 3 whose z-coordinate is 3 (x and y can each be any value). This is the
horizontal plane that is parallel to the xy-plane and three units above it as in Figure 7(a).
_2
3
z
0
y
z
z
y
x
3
0
y
0
x 5
y
5
0
x
FIGURE 7
(a) z=3, a plane in R#
(b) y=5, a plane in R#
(c) y=5, a line in R@
(b) The equation y − 5 represents the set of all points in R 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).
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