Question

Find the position vector of a point R which divides the line segment joining points $P\left(\hat{i}+2\hat{j}+\hat{k}\right)\mathrm{and}Q\left(-\hat{i}+\hat{j}+\hat{k}\right)$ in the ratio 2:1.
(i) internally
(ii) externally

Answer 1

(i) Given: R divides the line segment joining the points $P\left(\hat{i}+2\hat{j}+\hat{k}\right),Q\left(-\hat{i}+\hat{j}+\hat{k}\right)$ in the ratio 2 : 1 internally.
Therefore. position vector of R = $\frac{2\left(-\hat{i}+\hat{j}+\hat{k}\right)+1\left(\hat{i}+2\hat{j}+\hat{k}\right)}{2+1}$
= $\frac{1}{3}\left(-\hat{i}+4\hat{j}+3\hat{k}\right)$


(ii) Given: R divides the line segment joining the points $P\left(\hat{i}+2\hat{j}+\hat{k}\right),Q\left(-\hat{i}+\hat{j}+\hat{k}\right)$ in the ratio 2 : 1 externally.
Therefore. position vector of R = $\frac{2\left(-\hat{i}+\hat{j}+\hat{k}\right)-1\left(\hat{i}+2\hat{j}+\hat{k}\right)}{2-1}$
= $-3\hat{i}+\hat{k}$