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= Solution We have
2
|
|
a
b
 
r
r
 = (
) (
)
a
b
a
b
−
⋅
−
r
r
r
r
= a a
a b
b a
b b
−
⋅
−
⋅
+
⋅
r
r
r r
r r
r
r
© NCERT
not to be republished
 MATHEMATICS
446
B
C
A
a
b
+
a
b
=
2
2
|
|
2(
) |
|
a
a b
b
−
⋅
+
r
r
r
r
=
2
2
(2)
2(4)
(3)
−
+
Therefore
|
|
a
−b
r
r
 =
5
Example 18 If ar  is a unit vector and (
) (
)
8
x
a
x
a
−
⋅
+
=
r
r
r
r
, then find |
|
xr