Question

$ABCD$ is a quadrilateral. Prove that sum of the vectors $\overrightarrow{BA},\overrightarrow{BC},\overrightarrow{CD}$ and $\overrightarrow{DA}$ = $2\overrightarrow{AB}$

Answer 1

Let $ABCD$ be a quadrilateral

$\overrightarrow{BA}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DA}$

$=\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c}-\overrightarrow{b}+\overrightarrow{d}-\overrightarrow{c}+\overrightarrow{a}-\overrightarrow{d}$

$=2(\overrightarrow{a}-\overrightarrow{b})$

$=2\overrightarrow{AB}$

$\therefore \overrightarrow{BA}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DA}=2\overrightarrow{AB}$