© NCERT
not to be republished
 MATHEMATICS
456
Example 27 If 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
, 2
5 , 3
2
3   and  
6
 
 
 
 
 
 
i
j
k
i
j
i
j
k
i
j
k  are the position
vectors of points A, B, C and D respectively, then find the angle between AB
uuur
 and
CD
uuur Deduce that AB
uuur
 and CD
uuur
 are collinear Solution Note that if θ is the angle between AB and CD, then θ is also the angle
between AB  and  CD
uuur
uuur Now
AB
uuur
 = Position vector of B – Position vector of A
=
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(2
5 )
(
)
4
i
j
i
j
k
i
j
k
+
−
+
+
= +
−
Therefore
| AB|
uuur  =
2
2
2
(1)
(4)
( 1)
3 2
+
+ −
=
Similarly
CD
uuur  =
ˆ
ˆ
ˆ
2
8
2   and  |CD | 6 2
i
j
k
−
−
+
=
uuur
Thus
cos θ =
AB CD
|AB||CD|
 
uuur uuur
uuur uuur
=
1( 2)
4( 8)
( 1)(2)
36
1
36
(3 2)(6 2)
−
+
−
+ −
=−
= −
Since 0 ≤ θ ≤ π, it follows that θ = π