Explanation

(d)
$ABCDEF$ is a regular hexagon. We know from the hexagon that $\overrightarrow{\mathbf{AD}}$ is parallel to $\overrightarrow{\mathbf{BC}}$.
$\Rightarrow \ \overrightarrow{\mathbf{AD}}\mathbf{=}2\overrightarrow{\mathbf{BC}}$
Similarly, $\overrightarrow{\mathbf{EB}}$ is a parallel to $\overrightarrow{\mathbf{FA}}$
[image:10_image13.png]
$\Rightarrow \ \overrightarrow{\mathbf{EB}}\mathbf{=}2\overrightarrow{\mathbf{FA}}$
and $\overrightarrow{\mathbf{FC}}$ is parallel to $\overrightarrow{\mathbf{AB}}$.
$\Rightarrow \ \overrightarrow{\mathbf{FC}}\mathbf{=}2\overrightarrow{\mathbf{AB}}$
Thus, $\overrightarrow{\mathbf{AD}}\mathbf{+}\overrightarrow{\mathbf{EB}}\mathbf{+}\overrightarrow{\mathbf{FC}}\mathbf{=}2\overrightarrow{\mathbf{BC}}\mathbf{+}2\overrightarrow{\mathbf{FA}}\mathbf{+}2\overrightarrow{\mathbf{AB}}$
$= 2\left( \overrightarrow{\mathbf{FA}}\mathbf{+}\overrightarrow{\mathbf{AB}}\mathbf{+ \ }\overrightarrow{\mathbf{BC}}\ \right)$
$= 2\left( \overrightarrow{\mathbf{FC}}\ \right) = 2\left( 2\overrightarrow{\mathbf{AB}}\ \right) = 4\ \overrightarrow{\mathbf{AB}}$