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# The position vector of the point which divides the join of points $2\overrightarrow{a}-3\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$ in the ratio 3:1 is

$(A)\;3\overrightarrow{a}+2\overrightarrow{b}\quad(B)\;\frac{7\overrightarrow{a}+8\overrightarrow{b}}{4}\quad(C)\;\frac{3\overrightarrow{a}}{4}\quad(D)\;\frac{5\overrightarrow{a}}{4}$

Toolbox:
• $\overrightarrow r=\large\frac{m \overrightarrow b+ n \overrightarrow a}{m+n}$ where $\overrightarrow r$ divides $\overrightarrow {AB}$ internals in the ratio $m:n$
Let $\overrightarrow {OP}= 2 \overrightarrow a-3 \overrightarrow b$ and $\overrightarrow {OQ}= \overrightarrow a+ \overrightarrow b$
Given ratio is $3:1$
Let R be the point which divides line PQ in the ratio $3:1$
We know the position vector which divides the join of the points P and Q, whose position vector are $\overrightarrow {OP}$ and $\overrightarrow {OQ}$ in the ratio $m:n$ is given by
$\overrightarrow {OR}=\large\frac{m \overrightarrow {OQ}+n \overrightarrow {OP}}{m+n}$
Here $\overrightarrow {OP}=2 \overrightarrow a-3 \overrightarrow b$
$\qquad \overrightarrow {OQ}=\overrightarrow a+\overrightarrow b$
$\qquad m=3$ and $n=1$
Therefore $\overrightarrow {OR}=\large\frac{3(\overrightarrow a+\overrightarrow b)+1(2 \overrightarrow a-3 \overrightarrow b)}{3+1}$
$\qquad\qquad=\large\frac{5 \overrightarrow a}{4}$
Hence the correct option is $D$