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# L and M are two points with position vectors $2\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{a}+2\overrightarrow{b}$ respectively.Write the position vector of a point N which divides the line segment LM in the ratio 2 : 1 externally.

Let $\overrightarrow{OL}=2\overrightarrow{a}-\overrightarrow{b}$
$\overrightarrow{OM}=\overrightarrow{a}+2\overrightarrow{b} \qquad ratio=2:1$
$\overrightarrow{LM}=\large\frac{2(\overrightarrow{OM})-1(\overrightarrow{OL})}{2-1}$
$=\large\frac{2(\overrightarrow{a}+2\overrightarrow{b})-1(2\overrightarrow{a}-\overrightarrow{b})}{1}$
$=\large\frac{2\overrightarrow{a}+4\overrightarrow{b}-2 \overrightarrow{a}+ \overrightarrow{b}}{1}$
$=5\overrightarrow{b}$