And therefore, from Case I, the
midpoint R of  PQ
uuur
, will have its position vector as
OR
uuur  =
a2
b
+
r
r
Example 11 Consider two points P and Q with position vectors OP
3
2
a
b
=
−
uuur
r
r
 and
OQ
a
b
 
 
uuur
r
r Find the position vector of a point R which divides the line joining P and Q
in the ratio 2:1, (i) internally, and (ii) externally Solution
(i)
The position vector of the point R dividing the join of P and Q internally in the
ratio 2:1 is
OR
uuur  = 2(
)
(3
2 )
5
2
1
3
a
b
a
b
a
+
+
−
=
+
r
r
r
r
r
(ii)
The position vector of the point R dividing the join of P and Q externally in the
ratio 2:1 is
OR
uuur  = 2(
)
(3
2 )
4
2
1
a
b
a
b
b
a
+
−
−
=
−
−
r
r
r
r
r
r
Example 12 Show that the points 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
A(2
), B(
3
5 ), C(3
4
4 )
i
j
k
i
j
k
i
j
k
 
 
 
 
 
 
 are
the vertices of a right angled triangle Solution We have
 AB
uuur  =
ˆ
ˆ
ˆ
(1
2)
( 3 1)
( 5 1)
i
j
k
−
+ − +
+ − −
ˆ
ˆ
2ˆ
6
i
j
k
  
 
 
  BC
uuur  =
ˆ
ˆ
ˆ
(3 1)
( 4
3)
( 4
5)
i
j
k
−
+ − +
+ − +
ˆ
ˆ
ˆ
2i
j
k
=
−
+
and
CA
uuur
 =
ˆ
ˆ
ˆ
(2
3)
( 1
4)
(1
4)
i
j
k
−
+ − +
+
+
 
ˆ
ˆ
3ˆ
5
i
j
k
= − +
+
Fig 10