question_answer

Find the position vector of a point R which divides the line joining the points \[P(\hat{i}\,+2\hat{j}\,-\hat{k})\] and \[Q(-\hat{i}\,+\hat{j}\,+\hat{k})\] in the ratio 2 : 1 (i) Internally.                  (ii) externally.

Answer:

Given, \[\overrightarrow{OP}=\hat{i}+2\hat{j}\,-\hat{k}\] and \[\overrightarrow{OQ}=-\hat{i}+\hat{j}\,+\hat{k}\] (i) Let R divides PQ internally in the ratio 2 : 1. Then, Position vector of R \[=\frac{2(-\hat{i}+\hat{j}\,+\hat{k})+1(\hat{i}+2\hat{j}\,-\hat{k})}{2+1}=\frac{-\hat{i}+4\hat{j}\,+\hat{k}}{3}\]            (ii)  Let R divides PQ externally in the ratio 2 : 1. Then, position vector of R \[=\frac{2(-\hat{i}+\hat{j}\,+\hat{k})-1(\hat{i}+2\hat{j}\,-\hat{k})}{2-1}=-\,3\hat{i}+3\hat{k}\]