Find the position vector of a point R which divides the line joining the two points P and Q with position vectors and respectively in the ratio 1 : 2 internally and externally.

Let the position vectors of points P, Q and R be, and respectively.

Given and

(i) R divides PQ internally in the ratio 1:2

Recall the position vector of point P which divides AB, the line joining points A and B with position vectors and respectively, internally in the ratio m : n is

Here, m = 1 and n = 2.

We have and

Thus, the position vector of point R is .

(ii) R divides PQ externally in the ratio 1:2

Recall the position vector of point P which divides AB, the line joining points A and B with position vectors and respectively, externally in the ratio m : n is

Here, m = 1 and n = 2.

We have and

Thus, the position vector of point R is .