Question

If $ABCDEF$ is a regular hexagon and $\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}=\lambda \overrightarrow{AD}$, then $\lambda =$

Answer 1

In$\Delta ACD$,

$AD=AC+CD$

In $\Delta ADE$,

$AD=AE+ED$

Add both the equations,

$2AD=AC+CD+AE+ED$ (1)

In a regular hexagon, the opposite sides are equal, so,

$AF=CD$ and $ED=AB$

Therefore, equation (1) becomes,

$2AD=AC+AF+AE+AB$

Add AD to both sides of the above equation,

$3AD=AC+AF+AE+AB+AD$

Therefore, the value of $\lambda$ is 3.