Further,
if ar  is a unit vector, then it may be expressed in terms of its direction cosines as
ˆ
ˆ
ˆ
cos
cos
cos
a
i
j
k
=
α +
β +
γ
r
Example 13 Find the angle between two vectors  and 
a
rb
r
 with magnitudes 1 and 2
respectively and  when  
a b1
⋅
=
rr Solution Given 
1,|
|
1and | |
2
a b
a
b
 
 
 
r
r
r
r We have
1
1 1
cos
cos
2
3
|
||
a b|
a b
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
rr
r
r
© NCERT
not to be republished
VECTOR ALGEBRA
445
Example 14 Find angle ‘θ’ between the vectors 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
   and  
a
i
j
k
b
i
j
k
= +
−
= −
+
r
r Solution The angle θ between two vectors 
aand 
rb
r
  is given by
cosθ = |
||
a b|
a b
⋅ r
r
r
r
Now
a b
⋅
rr
 =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
) (
)
1 1 1
1
i
j
k
i
j
k
+
−
⋅
−
+
= − − = −