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
+
−
⋅
−
+
= − − = − Therefore, we have
  cosθ =
31
−
hence the required angle is
θ =
1
1
cos
3
 
 
 
  
Example 15 If 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
5
3   and  
3
5
a
i
j
k
b
i
j
k
=
−
−
=
+
−
r
r
, then show that the vectors
 and  
a
b
a
b
+
−
r
r
r
r
 are perpendicular Solution We know that two nonzero vectors are perpendicular if their scalar product
is zero