Find the position vector of a point C which divides the line segment joining A and B, whose position vectors are $2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}},$externally in theratio 1 : 2. Also, show that A is the mid-point of theline segment BC.

Given $\overrightarrow{\mathrm{OA}}=2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\mathrm{and} \overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$

Also, it is given that C is the point which divides the line joining A and B externally in the ratio 1:2. Then, by using section formula of external division, we get

$$\begin{gathered} \overrightarrow{\mathrm{OC}}=\frac{2\overrightarrow{\mathrm{OA}}-\overrightarrow{\mathrm{OB}}}{2-1} \\ \overrightarrow{\mathrm{OC}}=\frac{2(2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})-1(\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}})}{1} \left[\mathrm{from} \mathrm{Eq}.\left(\mathrm{i}\right)\right] \\ =4\overrightarrow{\mathrm{a}}+2\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}+3\overrightarrow{\mathrm{b}} \\ =3\overrightarrow{\mathrm{a}}+5\overrightarrow{\mathrm{b}} \dots \left(\mathrm{ii}\right) \end{gathered}$$

Now, we have to show that A is the mid-point of BCi.e. to show

$$\begin{gathered} \overrightarrow{\mathrm{OA}}=\frac{\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}}{2} \\ \mathrm{Consider},\frac{\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}}{2}=\frac{\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}+3\overrightarrow{\mathrm{a}}+5\overrightarrow{\mathrm{b}}}{2} \\ \left[\mathrm{from} \mathrm{Eqs}. \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right)\right] \\ =\frac{4\overrightarrow{\mathrm{a}}+2\overrightarrow{\mathrm{b}}}{2\mathrm{from} \mathrm{Eq}.\left(\mathrm{i}\right)}=2\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{OA}} \left[\mathrm{from} \mathrm{Eq}. \left(\mathrm{i}\right)\right] \\ \mathrm{Thus}, \frac{\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}}{2}=\overrightarrow{\mathrm{OA}} \end{gathered}$$

Hence, A is mid-point of line segment BC.