11.1 Three-Dimensional Coordinate Systems
In order to represent points in space, we rst choose a xed point O (the origin)
and three directed lines through O that are perpendicular to each other, called the
coordinate axes and labeled the x-axis, y -axis, and z -axis.
Their directions are determined by the right-hand rule: If you curl the ngers of
your right hand around the z -axis in the direction of a 90◦ counterclockwise rotation
from the positive x-axis to the positive y -axis, then your thumb points in the positive
direction of the z -axis.
The three coordinate axes determine the three coordinate planes. The xy -plane
is the plane that contains the x- and y -axes; the yz -plane contains the y - and z -axes;
the xz -plane contains the x- and z -axes.
If P is any point in space, let a be the (directed) distance from the yz -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 P by the ordered triple (a, b, c) 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.
If we drop a perpendicular from a point P (a, b, c) to the xy -plane, we get a point with
coordinates (a, b, 0) called the projection of P onto the xy -plane. Similarly, (0, b, c)
and (a, 0, c) are the projections of P onto the yz -plane and xz -plane, respectively.
Let R3 = {(x, y, z)|x, y, z ∈ R} be the set of all ordered triples of real numbers. We
have given a one-to-one correspondence between points P in space and ordered triples
(a, b, c) in R3 . It is called a three dimensional rectangular coordinate system.
Example 1. Describe the graph of z = 0 in R3 .
Cylindrical surfaces
Note that in R2 , the graph of the equation involving x and y is a curve. In R3 , an
equation in x, y, z represents a surface.
Example 2. Sketch the graph of x2 + y 2 = 1 in R2 and R3 .
An equation that contains only two of the variables
surface in
R
3
x, y, z
represents a cylindrical
. How to graph cylindrical surface:
1. graph the equation in the coordinate plane of the two variables that appear in the
given equation;
2. translate that graph parallel to the axis of the missing variable.
Example 3. Sketch the graph of x = y in R3 .
Example 4. Sketch the graph of (x + 2)2 + (z − 4)2 = 1 in R3 .
Example 5. Sketch the graph of z = y 2 in R3 .
Example 6. Sketch the graph of x2 + y 2 = 4, z = 5.
Distance formula in
R3
The distance |P1 P2 | between the points P1 (x1 , y1 , z1 ) and P2 (x2 , y2 , z2 ) is
|P1 P2 | =
p
(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
Equation of a sphere
A sphere is the set of all points whose distance from its center is equal to the radius.
Thus an equation of a sphere with center C(x0 , y0 , z0 ) and radius r is given by
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r2
Example 7. Find the center and radius of the sphere x2 + y 2 + z 2 = 6x − 4y + 10z .
Regions in
R3
Example 8. Describe the set of all points in R3 whose coordinates satisfy the inequality:
x2 + y 2 ≤ 4
Example 9. Describe the following region in R3 : { (x, y, z) | x2 + y 2 + z 2 < 4 }