Example 6 Find unit vector in the direction of vector 
ˆ
ˆ
ˆ
2
3
a
i
j
k
=
+
+
r
Solution The unit vector in the direction of a vector ar  is given by 
1
ˆ
|
|
a
=aa
rr Now
|
|
ar  =
2
2
2
2
3
1
14
+
+
=
Therefore
1
ˆ
ˆ
ˆ
ˆ
(2
3
)
14
a
i
j
k
=
+
+
 =
2
3
1
ˆ
ˆ
ˆ
14
14
14
i
j
k
+
+
Example 7 Find a vector in the direction of vector 
ˆ
2ˆ
a
i
j
r= −
 that has magnitude
7 units Solution The unit vector in the direction of the given vector ar  is
1
ˆ
|
|
a
=aa
rr
 =
1
1
2
ˆ
ˆ
ˆ
ˆ
(
2 )
5
5
5
i
j
i
j
−
=
−
Therefore, the vector having magnitude equal to 7 and in the direction of ar is
7a
∧  =
1
2
7
5
5
i
j
∧
∧
⎛
⎞
−
⎜
⎟
⎝
⎠
 = 
7
14
ˆ
ˆ
5
5
i
j
−
Example 8 Find the unit vector in the direction of the sum of the vectors,
ˆ
ˆ
ˆ
2
2 – 5
a
i
j
k
=
+
r
 and 
ˆ
ˆ
ˆ
2
3
b
i
j
k
=
+
+
r Solution The sum of the given vectors is
ˆ
ˆ
ˆ
(
, say) = 4
3
2
 
 
 
 
r
r
r
a
b
c
i
j
k
and
|
|
cr  =
2
2
2
4
3
( 2)
29
+
+ −
=
© NCERT
not to be republished
VECTOR ALGEBRA
437
Thus, the required unit vector is
1
1
4
3
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(4
3
2 )
|
|
29
29
29
29
c
c
i
j
k
i
j
k
=c
=
+
−
=
+
−
r
r
Example 9  Write the direction ratio’s of the vector 
ˆ
ˆ
ˆ
2
a
i
j
k
= +
−
r
 and hence calculate
its direction cosines