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# Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $(2\overrightarrow a + \overrightarrow b)$ and $(\overrightarrow a - 3\overrightarrow b)$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ

Toolbox:
• The position vector of a point which divides the line in the ratio $m:n$ externally is given by $\large\frac{m\overrightarrow a-n\overrightarrow b}{m-n}$
Step 1:
Let the position vector of $P$ be $\overrightarrow {OP}=2\overrightarrow a+\overrightarrow b$
Position vector of $Q$ be $\overrightarrow {OQ}=2\overrightarrow a-3\overrightarrow b$
Now it is given the point $R$ divides the line $PQ$ externally in the ratio $1:2$.
Step 2:
Therefore position vector of $R$ = $\large\frac{m\overrightarrow{OQ}-n\overrightarrow{OP}}{m-n}$
$\qquad\qquad\qquad\qquad\quad\quad\;\;=\large\frac{1.(\overrightarrow a-3\overrightarrow b)-2(2\overrightarrow a+\overrightarrow b)}{1-2}$
$\qquad\qquad\qquad\qquad\quad\quad\;\;=\large\frac{-3\overrightarrow a-5\overrightarrow b}{-1}=$$3\overrightarrow a+5\overrightarrow b$
Step 3:
Now the mid point of $RQ$ =$\large\frac{\overrightarrow{OR}+\overrightarrow{OQ}}{2}$
$\qquad\qquad\qquad\qquad\quad=\large\frac{(3\overrightarrow a+5\overrightarrow b)+(\overrightarrow a-3\overrightarrow b)}{2}$
$\qquad\qquad\qquad\qquad\quad=\large\frac{4\overrightarrow a+2\overrightarrow b}{2}$
$\qquad\qquad\qquad\qquad\quad=2\overrightarrow a+\overrightarrow b$
This is nothing but the position vector of point $P$
Step 4:
Hence $P$ is the mid point of $RQ$ and the point R is $3\overrightarrow a+5\overrightarrow b$