**Toolbox:**

- Given a point $R$ that divides a line $PQ$ externally in the ratio $m:n$, $\large \frac{P}{Q} = \frac {m}{n}$, the position vector of the point R is given by $\overrightarrow{OR}=\large \frac{m.\overrightarrow{OQ}-n.\overrightarrow{OP}}{m-n}$

Given the point R divides line PQ externally in the ratio 2 : 1, where the position vector of the two points P and Q are given by $\overrightarrow{OP}=3\overrightarrow{a}-2\overrightarrow{b}$ and $\overrightarrow{OQ}=\overrightarrow{a}+\overrightarrow{b}$

Given a point $R$ that divides a line $PQ$ externally in the ratio $m:n$, $\large \frac{P}{Q} = \frac {m}{n}$, the position vector of the point R is given by $\overrightarrow{OR}=\large \frac{m.\overrightarrow{OQ}-n.\overrightarrow{OP}}{m-n}$

$\Rightarrow$ $\overrightarrow{OR} =\Large \frac{2(\overrightarrow{a}+\overrightarrow{b})-1(3\overrightarrow{a}-2\overrightarrow{b})}{2-1}$

$\Rightarrow$ $\overrightarrow{OR} =\Large \frac{(2\overrightarrow{a}+2\overrightarrow{b}-3\overrightarrow{a}+2\overrightarrow{b})}{1}$

$\Rightarrow$ $\overrightarrow{OR} =4\overrightarrow{b}-\overrightarrow{a}$.