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

विकल्प

• \[\left[ \vec{a} \vec{b} \vec{c} \right]\]

• \[2\left[ \vec{a} \vec{b} \vec{c} \right]\]

• \[\left[ \vec{a} \vec{b} \vec{c} \right]^2\]

• none of these

Answer 1

none of these

We have 

\[\left( \vec{a} - \vec{b} \right) . \left[ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right]\]

\[ = \left( \vec{a} - \vec{b} \right) . \left[ \left( \vec{b} - \vec{c} \right) \times c - \left( \vec{b} - \vec{c} \right) \times \vec{a} \right] \]

\[ = \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{c} - \vec{c} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right) \]

\[ = \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{c} - 0 - \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right) \]

\[ = \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{c} \right) - \left( \vec{a} - \vec{b} \right) . \left( \vec{b} \times \vec{a} \right) + \left( \vec{a} - \vec{b} \right) . \left( \vec{c} \times \vec{a} \right) \hspace{0.167em} \]

\[ = \vec{a} . \left( \vec{b} \times \vec{c} \right) - \vec{b .} \left( \vec{b} \times \vec{c} \right) - \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{b} . \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{c} \times \vec{a} \right) - \vec{b} . \left( \vec{c} \times \vec{a} \right)\]

\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - 0 - 0 + 0 + 0 - \left[ \vec{b} \vec{c} \vec{a} \right] \left( \because \left[ \vec{b} \vec{b} \vec{c} \right] = \left[ \vec{a} \vec{b} \vec{a} \right] = \left[ \vec{b} \vec{b} \vec{a} \right] = 0 \right)\]

\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{c} \right] \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left[ \vec{b} \vec{c} \vec{a} \right] = \left[ \vec{c} \vec{a} \vec{b} \right] \right)\]

\[ = 0\]