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Question
For any three vectors \[\vec{a,} \vec{b,} \vec{c}\] the expression \[\left( \vec{a} - \vec{b} \right) . \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\}\] equals
Options
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[2\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[\left[ \vec{a} \vec{b} \vec{c} \right]^2\]
none of these
Solution
none of these
We have
\[\left( \vec{a} - \vec{b} \right) . \left[ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right]\]
\[ = \left( \vec{a} - \vec{b} \right) . \left[ \left( \vec{b} - \vec{c} \right) \times c - \left( \vec{b} - \vec{c} \right) \times \vec{a} \right] \]
\[ = \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{c} - \vec{c} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right) \]
\[ = \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{c} - 0 - \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right) \]
\[ = \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{c} \right) - \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{a} \right) + \left( \vec{a} - \vec{b} \right) . \left( \vec{c} \times \vec{a} \right) \hspace{0.167em} \]
\[ = \vec{a} . \left( \vec{b} \times \vec{c} \right) - \vec{b .} \left( \vec{b} \times \vec{c} \right) - \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{b} . \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{c} \times \vec{a} \right) - \vec{b} . \left( \vec{c} \times \vec{a} \right)\]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - 0 - 0 + 0 + 0 - \left[ \vec{b} \vec{c} \vec{a} \right] \left( \because \left[ \vec{b} \vec{b} \vec{c} \right] = \left[ \vec{a} \vec{b} \vec{a} \right] = \left[ \vec{b} \vec{b} \vec{a} \right] = 0 \right)\]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{c} \right] \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left[ \vec{b} \vec{c} \vec{a} \right] = \left[ \vec{c} \vec{a} \vec{b} \right] \right)\]
\[ = 0\]
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