of point Q are (3,5,2). Points are denoted as ordered triples in the form
(x,y,z), where the first number represents the value on the x axis, the second
number represents the value on the y axis, and the third number represents
the value on the z axis. The word ‘‘ordered’’ means that the order, or
sequence, in which the numbers are listed is important. The ordered triple
(1,2,3) is not the same as any of the ordered triples (1,3,2), (2,1,3), (2,3,1),
(3,1,2), or (3,2,1), even though all of the triples contain the same three
numbers.
In an ordered triple, there are no spaces after the commas, as there are in
the notation of a set or sequence.
VARIABLES AND ORIGIN
In Cartesian three-space, there are usually two independent-variable coordi-
nate axes and one dependent-variable axis. The x and y axes represent inde-
pendent variables; the z axis represents a dependent variable whose value is
affected by both the x and the y values.
In some scenarios, two of the variables are dependent and only one is
independent. Most often, the independent variable in such cases is x. Rarely,
you’ll come across a situation in which none of the values depends on either
of the other two, or when a correlation, but not a true relation, exists among
the values of two or all three of the variables. Plots of this sort usually look
like ‘‘swarms of points,’’ representing the results of observations, or values
predicted by some scientific theory. These graphs are known as scatter plots,
and are common in statistics.
DISTANCE BETWEEN POINTS
Suppose there are two different points P ¼ (x
0
,y
0
,z
0
) and Q ¼ (x
1
,y
1
,z
1
)in
Cartesian three-space. The distance d between these two points can be
found using this formula:
d ¼½ðx
1
x
0
Þ
2
þðy
1
y
0
Þ
2
þðz
1
z
0
Þ
2
1=2
PROBLEM 13-4
What is the distance between the points P ¼ (–5,–4,3) and Q ¼ (3,5,–2)
illustrated in Fig. 13-3? Express the answer rounded off to three decimal
places.
CHAPTER 13 Vectors and 3D 315