Question

The position vector of the point which divides the join of points $2\overrightarrow{a}-3\overrightarrow{b}\mathrm{and}\overrightarrow{a}+\overrightarrow{b}$in the ratio 3:1 is
$$\begin{gathered} \left(\mathrm{a}\right)\frac{1}{2}\left(8\overrightarrow{a}-2\overrightarrow{b}\right) \\ \left(\mathrm{b}\right)\frac{1}{4}\left(7\overrightarrow{a}-8\overrightarrow{b}\right) \\ \left(\mathrm{c}\right)\frac{3}{4}\overrightarrow{a} \\ \left(\mathrm{d}\right)\frac{5}{4}\overrightarrow{a} \end{gathered}$$

Answer 1

Given:
X and Y are two points with position vectors $2\overrightarrow{a}-3\overrightarrow{b}\mathrm{and}\overrightarrow{a}+\overrightarrow{b}$, respectively.
Z divides the line segment XY in the ratio 3 : 1


We know,
The position vector of R which divided P and Q internally in the ratio m : n is given by $\overrightarrow{OR}=\frac{m\overrightarrow{b}+n\overrightarrow{a}}{m+n}$, where position vector of P and Q are $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$, respectively.

Thus,
$$\begin{gathered} \overrightarrow{OZ}=\frac{3\left(\overrightarrow{a}+\overrightarrow{b}\right)+1\left(2\overrightarrow{a}-3\overrightarrow{b}\right)}{3+1} \\ \Rightarrow \overrightarrow{OZ}=\frac{3\overrightarrow{a}+3\overrightarrow{b}+2\overrightarrow{a}-3\overrightarrow{b}}{4} \\ \Rightarrow \overrightarrow{OZ}=\frac{5\overrightarrow{a}}{4} \end{gathered}$$

Hence, the correct option is (d).