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 a
b
a
b
i
j
k
i
j
k
+
⋅
−
=
+
−
⋅
−
+
=
−
−
=
r
r
r
r
Hence
  and  
a
b
a
b
+
−
r
r
r
r
 are perpendicular vectors Example 16 Find the projection of the vector 
ˆ
ˆ
ˆ
2
3
2
a
i
j
k
=
+
+
r
 on the vector
ˆ
ˆ
2ˆ
b
i
j
k
=
+
+
r Solution The projection of vector ar  on  the vector b
r
 is given by
1 (
)
|
|
a b
b
⋅
rr
r
 = 
2
2
2
(2 1
3 2
2 1)
10
5
6
3
6
(1)
(2)
(1)
× + ×
+
×
=
=
+
+
Example 17 Find |
|
a
b
−
r
r
, if two vectors 
aand
rb
r
 are such that |
|
2, |
|
3
a
b
 
r 
r
and 
4
a b
⋅
rr=