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If $\vec{a}$ and $\overrightarrow{\mathrm{b}}$ are position vectors of the points $\mathrm{A}$ and $\mathrm{B}$ respectively, then what is the position vector of a point $\mathrm{C}$ on $\mathrm{AB}$ produced such that $\overrightarrow{\mathrm{AC}}=\overrightarrow{2 \mathrm{AB}}$ ?
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The correct answer is:
$\quad 2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$
Let $\vec{c}$ be the position vector of point $C$ on AB produced. From the laws of vector addition,
$\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}$
$\Rightarrow \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$
and similarly in $\Delta \mathrm{AOC}, \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}$
$\Rightarrow \overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}$
As given, $\overrightarrow{\mathrm{AC}}=2 \overrightarrow{\mathrm{AB}}=2 \overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}$
$\Rightarrow 2 \mathrm{~b}-2 \mathrm{a}=\mathrm{c}-\mathrm{a}$
$\Rightarrow \overrightarrow{\mathrm{c}}=2 \overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{a}}=2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$
$\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}$
$\Rightarrow \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$
and similarly in $\Delta \mathrm{AOC}, \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}$
$\Rightarrow \overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}$
As given, $\overrightarrow{\mathrm{AC}}=2 \overrightarrow{\mathrm{AB}}=2 \overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}$
$\Rightarrow 2 \mathrm{~b}-2 \mathrm{a}=\mathrm{c}-\mathrm{a}$
$\Rightarrow \overrightarrow{\mathrm{c}}=2 \overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{a}}=2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$
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