Question

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors \(\vec{OP}\) = 2 \(\vec a\) + \(\vec b\) and \(\vec{OQ}=\vec a-2\vec b,\) respectively in the ratio 1 : 2 internally and externally.

Answer 1

Let the position vectors of points P, Q and R be \(\vec p,\vec q\) and \(\vec r\) respectively.

Given \(\vec p= \vec{OP}=2\vec a+\vec b\) and \(\vec q = \vec{OQ} = \vec a - 2\vec b\) 

(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 \(\vec a\)and \(\vec b\)respectively, internally in the ratio m : n is

Thus, the position vector of point R is \(\cfrac53\vec a\).

(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 \(\vec a\)and \(\vec b\)respectively, externally in the ratio m : n is

Thus, the position vector of point R is

\(3\vec a+4\vec b\).