Express in terms of single vector

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Can you guys give me a hint for this problem.

ABCDEF is a regular hexagon.

Express in terms of a single vector the sum of the vectors, $4\overrightarrow{AB}$, $2\overrightarrow{AC}$, $\overrightarrow{AD}$, $\overrightarrow{AE}$, and $5\overrightarrow{AF}$.

I was able to combine $4\overrightarrow{AB} + 4\overrightarrow{AF} = 4\overrightarrow{AO} = 2\overrightarrow{AD}$. I am a little stuck with the other vectors.

Thanks for all your help.

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1 Answer

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It will make things a whole lot easier when you notice that

$$\overrightarrow{AA}+\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{AE}=\overrightarrow{AC}+\overrightarrow{AF}=2\overrightarrow{AO}+\overrightarrow{OC}+\overrightarrow{OF}=2\overrightarrow{AO}\;,$$

and in particular

$$\overrightarrow{AA}+\overrightarrow{AD}+\overrightarrow{AB}+\overrightarrow{AE}+\overrightarrow{AC}+\overrightarrow{AF}=6\overrightarrow{AO}\;.$$

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