Question

Consider the regular hexagon $ABCDEF$ with centre at $O$ (origin).
$\overrightarrow{AD}+\overrightarrow{EB}+\overrightarrow{FC}$ is equal to

• A. $\overrightarrow{AB}$
• B. $3\overrightarrow{AB}$
• C. $4\overrightarrow{AAB}$
• D. none of these

Answer 1

Answer: (C)

Consider the regular hexagon $ABCDEF$ with centre at $O$ (origin).

$=\overrightarrow{AD}+\overrightarrow{EB}+\overrightarrow{FC}=2\overrightarrow{AO}+2\overrightarrow{OB}+2\overrightarrow{OC}$
$=2(\overrightarrow{AO}+\overrightarrow{OB})+2\overrightarrow{OC}=2\overrightarrow{AB}+2\overrightarrow{AB}$
$=(\because \overrightarrow{OC}=\overrightarrow{AB})=4\overrightarrow{AB}$
$=\overrightarrow{R}=\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}$
$=\overrightarrow{ED}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{CD}$
$=(\because \overrightarrow{AB}=\overrightarrow{ED}$ and $\overrightarrow{AF}=\overrightarrow{CD})$
$=(\overrightarrow{AC}+\overrightarrow{CD})+(\overrightarrow{AE}+\overrightarrow{ED})+\overrightarrow{AD}$
$=\overrightarrow{AD}+\overrightarrow{AD}+\overrightarrow{AD}=3\overrightarrow{AD}=6\overrightarrow{AO}$