question_answer

If ABCDEF is a regular hexagon and\[\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}=k\overrightarrow{AD}\], then find the value of k.

A) 2

B) 3

C) 4

D) 5

Correct Answer: B

Solution :

[b] \[\because \,\,\overrightarrow{AB}=\overrightarrow{ED}\] and \[\overrightarrow{AF}=\overrightarrow{CD}\], so \[\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}\] \[=\overrightarrow{ED}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{CD}\] \[=(\overrightarrow{AC}+\overrightarrow{CD})+(\overrightarrow{AE}+\overrightarrow{ED})+\overrightarrow{AD}\] \[=\overrightarrow{AD}+\overrightarrow{AD}+\overrightarrow{AD}+=3\overrightarrow{AD}\therefore k=3\]