Question

Find the position vector of a point $R$ which divides the line joining $A(--2,1,3)$ and $B(3,5,-2)$ in the ratio $2:1$
(i) Internally
(ii) externally.

Answer 1

$\overrightarrow{OA}=2\hat{i}+\hat{j}+3\hat{k}$

$\overrightarrow{OB}=3\hat{i}+5\hat{j}-2\hat{k}$

(i) internally

Position vector of R $=\frac{2\times \overrightarrow{OB}+1\times \overrightarrow{OA}}{2+1}$

$=\frac{2\times \overrightarrow{OB}+1\times \overrightarrow{OA}}{2+1}$

$\overrightarrow{OR}=\frac{2\times (3\hat{i}+5\hat{j}-2\hat{k})+1\times 2\hat{i}+\hat{j}+3\hat{k}}{2+1}$

$=\frac{6\hat{i}+10\hat{j}-4\hat{k}+2\hat{i}+\hat{j}+3\hat{k}}{3}$

$=\frac{8\hat{i}+11\hat{j}-\hat{k}}{3}$

The position vector of R dividing A and B internally is $\frac{8}{3}\hat{i}+\frac{11}{3}\hat{j}-\hat{k}$

(ii) externally

Position vector of R $=\frac{2\times \overrightarrow{OB}-1\times \overrightarrow{OA}}{2-1}$

$=\frac{2\times \overrightarrow{OB}-1\times \overrightarrow{OA}}{2-1}$

$\overrightarrow{OR}=\frac{2\times (3\hat{i}+5\hat{j}-2\hat{k})-1\times 2\hat{i}+\hat{j}+3\hat{k}}{2-1}$

$=\frac{6\hat{i}+10\hat{j}-4\hat{k}-2\hat{i}-\hat{j}-3\hat{k}}{1}$

$=4\hat{i}+9\hat{j}-7\hat{k}$

The position vector of R dividing A and B externally is $4\hat{i}+9\hat{j}-7\hat{k}$