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Question
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar vectors, then \[\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]\] equals
Options
0
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[2\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[- \left[ \vec{a} \vec{b} \vec{c} \right]\]
Solution
\[ - \left[ \vec{a} \vec{b} \vec{c} \right]\]
We have
\[\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]\]
\[ = \left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \vec{a} + \left( \vec{a} + \vec{b} \right) \times \vec{c} \right] \left(\text { By definition of cross poduct }\right)\]
\[ = \left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \vec{a} \times \vec{a} + \vec{b} \times \vec{a} + \vec{a} \times \vec{c} + \vec{b} \times \vec{c} \right]\]
\[ = \left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ 0 + \vec{b} \times \vec{a} + \vec{a} \times \vec{c} + \vec{b} \times \vec{c} \right]\]
\[ = \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{a} \times \vec{c} \right) + \vec{a} . \left( \vec{b} \times \vec{c} \right) + \vec{b} . \left( \vec{b} \times \vec{a} \right) + \vec{b} . \left( \vec{a} \times \vec{c} \right) + \vec{b} . \left( \vec{b} \times \vec{c} \right) + \vec{c} . \left( \vec{b} \times \vec{a} \right) + \vec{c} . \left( \vec{a} \times \vec{c} \right) + \vec{c} . \left( \vec{b} \times \vec{c} \right) \]
\[ = \left[ \vec{a} \vec{b} \vec{a} \right] + \left[ \vec{a} \vec{a} \vec{c} \right] + \left[ \vec{a} \vec{b} \vec{c} \right] + \left[ \vec{b} \vec{b} \vec{a} \right] + \left[ \vec{b} \vec{a} \vec{c} \right] + \left[ \vec{b} \vec{b} \vec{c} \right] + \left[ \vec{c} \vec{b} \vec{a} \right] + \left[ \vec{c} \vec{a} \vec{c} \right] + \left[ \vec{c} \vec{b} \vec{c} \right]\]
\[ = 0 + 0 + \left[ \vec{a} \vec{b} \vec{c} \right] + 0 + \left[ \vec{b} \vec{a} \vec{c} \right] + 0 + \left[ \vec{c} \vec{b} \vec{a} \right] + 0 + 0\]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{c} \right] \left( \because \left[ \vec{b} \vec{a} \vec{c} \right] = - \left[ \vec{c} \vec{a} \vec{b} \right], \left[ \vec{b} \vec{a} \vec{c} \right] = - \left[ \vec{a} \vec{b} \vec{c} \right] \right)\]
\[ = - \left[ \vec{a} \vec{b} \vec{c} \right]\]
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