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# $(\overrightarrow a+\overrightarrow b+\overrightarrow c).[(\overrightarrow a+\overrightarrow b)\times (\overrightarrow a+\overrightarrow c)]=?$

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## 1 Answer

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Toolbox:
• $\overrightarrow a\times\overrightarrow a=0$
• $\overrightarrow a\times\overrightarrow b=-(\overrightarrow b\times\overrightarrow a)$
• $[\overrightarrow a\:\overrightarrow a\:\overrightarrow c]=0$
• $[\overrightarrow a\:\overrightarrow c\:\overrightarrow b]=-[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$
$(\overrightarrow a+\overrightarrow b+\overrightarrow c).[(\overrightarrow a+\overrightarrow b)\times(\overrightarrow a+\overrightarrow c)]$
$=(\overrightarrow a+\overrightarrow b+\overrightarrow c).[\overrightarrow a\times \overrightarrow c+\overrightarrow b\times \overrightarrow a+\overrightarrow b\times\overrightarrow c]$
$=\overrightarrow a.(\overrightarrow b\times \overrightarrow c)+\overrightarrow b.(\overrightarrow a\times \overrightarrow c)+\overrightarrow c.(\overrightarrow b\times \overrightarrow a)$
$=[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]-[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]-[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$
$=-[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$
answered Nov 1, 2013

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