Vectors and shapes

Question

In a regular hexagon, $\vec{AB}$+$\vec{AC}$+$\vec{AD}$+$\vec{AE}$+$\vec{AF}$=k$\vec{AD}$, then what is the value of k?

Concepts I know, Triangle and parallelogram law of vector addition.

Answer 1

Hint: $$ (\vec{AB}+\vec{AE})+\vec{AD}+(\vec{AF}+\vec{AC}) $$ and remember that $\vec{AB}=\vec{ED}$ ...

Answer 2

Let $\vec{AB}=a$ and $\vec{AF}=b$. Then $\vec{AC}=b+2a$, $\vec{AD}=2a+2a$, $\vec{AE}=2b+a$. Hence $$\vec{AB}+\vec{AC}+\vec{AD}+\vec{AE}+\vec{AF}=6b+6a=3\vec{AD}.$$