Download 12.1 Three-Dimensional Coordinate Systems

Instructor: Longfei Li
Math 243 Lecture Notes
12.1 Three-Dimensional Coordinate Systems
Much of what we will do is similar to Calculus I, but on higher dimensions. We will deal with functions
with several variables instead of just a single variable as we did before. Previously, we mostly work in
the 2D space, now we will work in 3D space.
A point in a plane is represented as an ordered pair of real number (a, b); a is the x−coordinate, b is
the y−coordinate
In space, a point is represented as an ordered triple of real numbers (a, b, c); a is the x−coordinate, b
is the y−coordinate and c is the z−coordinate.
We need coordinate axes to represent points in space. x−, y−, z− axes follow the Right-Hand rule.
Axes divide the space into 8 parts, called octants. Recall in a plane coordinate axes divide the plane
into 4 parts, called quadrants.
Figure 1: Coordinate Axes
A point P in a space determines a rectangular box. From the box, we know the projection points of P
onto the coordinate axes and coordinate planes (eg: xy−, yz− planes).
Example1: The point (−4, 3, −5) determines a rectangular box as below. The projection onto the
xz−plane is (−4, 0, −5). The projection onto the y−axis is (0, 3, 0). The distance to the y−axis is
1
√
41.
Notation: R3 = {(x, y, z)|x, y, z ∈ R} is the set of all triples of real numbers.
3D rectangular coordinate system:
1−to−1
Any point P in space ⇐⇒ ordered triples (a, b, c) ∈ R3
In 2D, the graph of an equation involving x, y is a curve in R2 . In 3D, the graph of an equation
involving x, y, z is a surface in R3
Example 2:
y = 5 is a plane in R3 while it represents a line in R2 .
Remark: Context is important! Same equation represents different geometry in R2 and R3
Example 3:
x2 + y 2 = 1 is a cylinder in R3 , while it’s a unit circle with center at the origin in R2 .
Distance formula in 3D:
The distance |AB| between the points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ) is
p
|AB| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
2
Equation of a sphere:
(x − a)2 + (y − b2 ) + (z − c2 ) = r2
here P (a, b, c) is the center, r is the radius.
3