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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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    Thanks, drawing an accurate picture helped.2011-07-20

1 Answers 1

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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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    Thanks, Using the center of the hexagon as the orgin helped. I ended up with something like this, \begin{align} 4\overrightarrow{AB} + 2\overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{AE} + 5\overrightarrow{AF} &= 4\overrightarrow{AB} + (2\overrightarrow{AB} + 2\overrightarrow{AO}) + 2\overrightarrow{AO} + \overrightarrow{AF} + \overrightarrow{AO} + 5\overrightarrow{AF} \\ &= 6\overrightarrow{AB} + 5\overrightarrow{AO} + 6\overrightarrow{AF} \\ &= 6\overrightarrow{AO} + 5\overrightarrow{AO} \\ &= 11\overrightarrow{AO} \\ &= \dfrac{11}{2}\overrightarrow{AD} \end{align} 2011-07-20