10
CHAPTER
1V
ECTORS AND VECTOR-VALUED FUNCTIONS
Parametric Equations for a Line.
The parametric equations for
the line through the point
x
0
,y
0
,z
0
parallel to the vector
m
m
1
,m
2
3
are
x
m
1
t
x
0
,
y
m
2
t
y
0
,
z
m
3
t
z
0
,
where
t
.
Notice that unlike in the
2
d case, where the
slope-intercept form
y
mx
b
uniquely de-
termines a line, in
3
d neither the vector nor the parametric equations uniquely determine a
line. It is always possible to give different equations for the line by using a different “base
point”
P
0
x
0
0
0
.
Example 1.
Give a vector equation for the line through the point
4
,
0
,
2
parallel to the
vector
2
,
1
,
1
.
Solution.
We have been given everything we need. The vector equation
r
t
2
,
1
,
1
t
4
,
0
,
2
speci±es this line.
Example 2.
Give a vector equation for the line through the points
P
3
,
2
,
2
and
Q
0
,
1
,
3
.
Solution.
This line is parallel to the vector
PQ
,
0
3
,
1
2
,
3
2
,
3
,
1
,
1
,
so the line is given by
r
t
3
,
1
,
1
t
3
,
2
,
2
.
(Note that here we used the point
P
as our base point, but we could instead have chosen
Q
, or indeed, any other point on the line.)
Example 3.
Determine whether the point
P
3
,
1
,
4
lies on the line
r
t
0
,
2
,
2
t
3
,
5
,
9
.