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Question
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) =\]
Options
0
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[2\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
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Solution
\[ \left[ \vec{a} \vec{b} \vec{c} \right] \]
We have
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left[ \left( \vec{b} + \vec{c} \right) \times \vec{a} + \left( \vec{b} + \vec{c} \right) \times \vec{b} + \left( \vec{b} + \vec{c} \right) \times \vec{c} \right]\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} + \vec{b} \times \vec{b} + \vec{c} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{c} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} + 0 + \vec{c} \times \vec{b} + \vec{b} \times \vec{c} + 0 \right) \]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} - \vec{b} \times \vec{c} + \vec{b} \times \vec{c} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right)\]
\[ = \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{b} . \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{c} \times \vec{a} \right) + \vec{b} . \left( \vec{c} \times \vec{a} \right)\]
\[ = 0 + 0 + 0 + \left[ \vec{b} \vec{c} \vec{a} \right] \]
\[ = \left[ \vec{b} \vec{c} \vec{a} \right] \]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] \]
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