e 1
2
3 ˆ
ˆ
ˆ
a
a i
a j
a k
=
+
+
r
 and 
1
2
3 ˆ
ˆ
ˆ
b
b i
b j
b k
=
+
+
r
,
then the two vectors are collinear if and only if
1
2
3 ˆ
ˆ
ˆ
b i
b j
b k
+
+
 =
1
2
3 ˆ
ˆ
ˆ
(
)
a i
a j
a k
λ
+
+
⇔
1
2
3 ˆ
ˆ
ˆ
b i
b j
b k
+
+
 =
1
2
3 ˆ
ˆ
ˆ
(
)
(
)
(
)
a i
a
j
a k
λ
+ λ
+ λ
⇔
1
1
b
= λa
,
2
2
3
3
,
b
a
b
a
= λ
= λ
⇔
1
1
b
a  =
3
2
2
3
b
ab
=a
= λ
(ii)
If 
1
2
3 ˆ
ˆ
ˆ
a
a i
a j
a k
 
 
 
r
, then a1, a2, a3 are also called direction ratios of ar (iii)
In case if it is given that  l, m, n are direction cosines of a vector, then 
ˆ
ˆ
ˆ
li
mj
nk
+
+
= 
ˆ
ˆ
ˆ
(cos )
(cos )
(cos )
i
j
k
α
+
β
+
γ
 is the unit vector in the direction of that vector,
where α, β and γ are the angles which the vector makes with x, y and z axes
respectively Example 4 Find the values of x, y and z so that the vectors 
ˆ
ˆ
2ˆ
a
xi
j
zk
=
+
+
r
 and
ˆ
ˆ
ˆ
2
b
i
yj
k
=
+
+
r
 are equal