Question

Write the position vector of the point which divides the join of points with position vectors $3\overrightarrow{a}-2\overrightarrow{b}\mathrm{and}2\overrightarrow{a}+3\overrightarrow{b}$ in the ratio 2 : 1.

Answer 1

Suppose R be the point which divides the line joining the points with position vectors $3\overrightarrow{a}-2\overrightarrow{b}\mathrm{and}2\overrightarrow{a}+3\overrightarrow{b}$ in the ratio 2 : 1
And, $\overrightarrow{\mathrm{OA}}=3\overrightarrow{a}-2\overrightarrow{b}\mathrm{and}\overrightarrow{\mathrm{OB}}=2\overrightarrow{a}+3\overrightarrow{b}$
Here, m : n = 2 : 1
Therefore, position vector $\overrightarrow{\mathrm{OR}}$ is as follows:
$$\begin{gathered} \overrightarrow{\mathrm{OR}}=\frac{m\overrightarrow{\mathrm{OB}}+n\overrightarrow{\mathrm{OA}}}{m+n} \\ =\frac{2(2\overrightarrow{a}+3\overrightarrow{b})+1(3\overrightarrow{a}-2\overrightarrow{b})}{2+1} \\ =\frac{7\overrightarrow{a}+4\overrightarrow{b}}{3} \\ =\frac{7}{3}\overrightarrow{a}+\frac{4}{3}\overrightarrow{b} \end{gathered}$$