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# P and Q are two points with position vectors $3\overrightarrow{a}-2\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$ respectively.Write the position vector of a point R which divides the line segment PQ in the ratio 2 : 1 externally.

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• 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 $\overrightarrow{OR}=\large \frac{m.\overrightarrow{OQ}-n.\overrightarrow{OP}}{m-n}$
$\Rightarrow$ $\overrightarrow{OR} =\Large \frac{2(\overrightarrow{a}+\overrightarrow{b})-1(3\overrightarrow{a}-2\overrightarrow{b})}{2-1}$
$\Rightarrow$ $\overrightarrow{OR} =\Large \frac{(2\overrightarrow{a}+2\overrightarrow{b}-3\overrightarrow{a}+2\overrightarrow{b})}{1}$
$\Rightarrow$ $\overrightarrow{OR} =4\overrightarrow{b}-\overrightarrow{a}$.

edited Mar 21, 2013