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 Here
a
b
+
r
r
 =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(5
3 )
(
3
5 )
6
2
8
i
j
k
i
j
k
i
j
k
−
−
+
+
−
=
+
−
and
a
−b
r
r
 =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(5
3 )
(
3
5 )
4
4
2
i
j
k
i
j
k
i
j
k
−
−
−
+
−
=
−
+
So
      
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
) (
)
(6
2
8 ) (4
4
2 )
24
8 16
0