Find the position vectors of the points which divide the line joining the points $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$ and $3\overrightarrow{\mathrm{a}}-2\overrightarrow{\mathrm{b}}$ internally and externally in the ratio 2 : 3.

Let A and B be the given points with position vectors $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$ and $3\overrightarrow{a}-2\overrightarrow{b}$respectively. Let P and Q be the points dividing AB in the ratio 2 : 3 internally and externally respectively. Then,

Position vector of P $=\frac{3\left(2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}\right)+2\left(3\overrightarrow{\mathrm{a}}-2\overrightarrow{\mathrm{b}}\right)}{3+2}=\frac{12\overrightarrow{\mathrm{a}}}{5}-\frac{13\overrightarrow{\mathrm{b}}}{5}$

Position vector of $\mathrm{Q}=\frac{3\left(2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}\right)-2\left(3\overrightarrow{\mathrm{a}}-2\overrightarrow{\mathrm{b}}\right)}{3-2}=-5\overrightarrow{\mathrm{b}}$