Question

ABCD is a parallelogram with AC and BD as diagonals. Then, $\overrightarrow{AC}-\overrightarrow{BD}=$
(a) $4\overrightarrow{AB}$
(b) $3\overrightarrow{AB}$
(c) $2\overrightarrow{AB}$
(d) $\overrightarrow{AB}$

Answer 1

(c) $2\overrightarrow{AB}$
Given: $ABCD$, a parallelogram with diagonals $AC$ and $BD$. Then,
$$\begin{gathered} \overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC} \\ \overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BD} \\ \Rightarrow \overrightarrow{BD}=\overrightarrow{AD}-\overrightarrow{AB} \end{gathered}$$

∴ $\overrightarrow{AC}-\overrightarrow{BD}=\overrightarrow{AB}+\overrightarrow{BC}-\overrightarrow{AD}+\overrightarrow{AB}=2\overrightarrow{AB}$ [∵ $\overrightarrow{AD}=\overrightarrow{BC}$]