Solution Let  ar  and  b
r
 be the position vectors of the point A(– 1, 0, 2) and B(3, 4, 6) Then
ˆ
ˆ
2
a
i
=−+k
r
and
ˆ
ˆ
ˆ
3
4
6
b
i
j
k
= + +
r
Therefore
ˆ
ˆ
ˆ
4
4
4
b
a
i
j
k
−
=
+
+
r
r
Let rr  be the position vector of any point on the line Then the vector equation of
the line is
ˆ
ˆ
ˆ
ˆ
ˆ
2
(4
4
4 )
r
i
k
i
j
=−+ +λ + +k
r
Example 8 The Cartesian equation of a line is
3
5
6
2
4
2
x
y
z
+ − +
=
=
Find the vector equation for the line Solution Comparing the given equation with the standard form
1
1
1
x
x
y
y
z
z
a
b
c
−
−
−
=
=
We observe that
x1 = – 3, y1 = 5, z1 = – 6; a = 2, b = 4, c = 2