Let ABCDEF be a regular hexagon. If \vec{AD}=m\vec{BC} and \vec{CF}=n\vec{AB}, then what is mn equal to?

Explanation

In a regular hexagon, the major diagonals are parallel to and twice the length of the corresponding opposite sides. Thus, \vec{AD} = 2\vec{BC}, so m=2. Similarly, \vec{CF} is parallel to \vec{BA}, so \vec{CF} = 2\vec{BA} = -2\vec{AB}, giving n=-2. Therefore, mn = 2 \times (-2) = -4.

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