Question

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

Answer 1

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


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

Hence, the correct option is (d).