$ABCDEF$ is a regular hexagon. What is $\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{AE} + \overrightarrow{AF}$?

Explanation

1. 01

   Opposite sides of a regular hexagon are equal and parallel: $\overrightarrow{AB} = \overrightarrow{ED}$ and $\overrightarrow{AF} = \overrightarrow{CD}$.

2. 02

   By the triangle law, $\overrightarrow{AE} + \overrightarrow{ED} = \overrightarrow{AD}$ and $\overrightarrow{AC} + \overrightarrow{CD} = \overrightarrow{AD}$.

3. 03

   So the sum is $\overrightarrow{AD} + \overrightarrow{AD} + \overrightarrow{AD} = 3\overrightarrow{AD}$.