We leave it to the reader as an exercise to verify
that the position vector of the point R which divides
the line segment PQ externally in the ratio
m : n 
PR
 i e QR
 
 
 
 
 
 
nm
 is given by
OR
uuur  =
mb
na
m
−n
−
r
r
Remark If R is the midpoint of PQ , then m = n 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