Question

If ABCDEF is a regular hexagon with $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{BC}=\overrightarrow{b}$,then $\overrightarrow{CE}$

• A. $\overrightarrow{v}-\overrightarrow{b}$
• B. $-\overrightarrow{b}$
• C. $\overrightarrow{b}-2\overrightarrow{a}$
• D. $Noneofthese$

Answer 1

The correct option is C $\overrightarrow{b}-2\overrightarrow{a}$

In $\Delta ABC,\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$
$\Rightarrow \overrightarrow{AC}=\overrightarrow{a}+\overrightarrow{b}.....\left(i\right)$
AD is parallel to BC and AD = 2BC, therefore $\overrightarrow{AD}=2\overrightarrow{b}$.

In $\Delta ACD,\overrightarrow{AC}+\overrightarrow{CD}=\overrightarrow{AD}$
$\Rightarrow \overrightarrow{CD}=2\overrightarrow{b}-(\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{b}-\overrightarrow{a}$
$\therefore$ In $\Delta CDE,\overrightarrow{CE}=\overrightarrow{CD}+\overrightarrow{DE}=\overrightarrow{b}-\overrightarrow{a}+(-\overrightarrow{a})=\overrightarrow{b}-2\overrightarrow{a}$