Find the position vector of a point R which divides the line segment joining P and Q whose position vectors are $2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$, externally in the ratio 1 : 2. Also, show that P is the mid-point of the line segment RQ.

It is given that Rr divides PQ externally in the ratio 1 : 2.

$\therefore$ The position vector of $\mathrm{R}=\frac{1\times \left(\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}\right)-2\left(2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right)}{1-2}=3\overrightarrow{\mathrm{a}}+5\overrightarrow{\mathrm{b}}$

Now, $\frac{\mathrm{Position}\mathrm{vector}\mathrm{of}\mathrm{R}+\mathrm{Position}\mathrm{vector}\mathrm{of}\mathrm{Q}}{2}=\frac{3\overrightarrow{\mathrm{a}}+5\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}}{2}$$=2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ Position vector of P

Hence, P is the mid-point of RQ.