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Question
The value of \[\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right], \text { where } \left| \vec{a} \right| = 1, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 3, \text { is }\]
Options
0
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6
none of these
Solution
We have
\[\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right]\]
\[ = \left( \left( \vec{a} - \vec{b} \right) \times \left( \vec{b} - \vec{c} \right) \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \left( \text { By definition of scalar triple product } \right)\]
\[ = \left( \left( \vec{a} - \vec{b} \right) \times \vec{b} - \left( \vec{a} - \vec{b} \right) \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \]
\[ = \left( \vec{a} \times \vec{b} - \vec{b} \times \vec{b} - \vec{a} \times \vec{c} + \vec{b} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \]
\[ = \left( \vec{a} \times \vec{b} - 0 - \vec{a} \times \vec{c} + \vec{b} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \]
\[ = \left( \vec{a} \times \vec{b} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} - \left( \vec{a} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} + \left( \vec{b} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \]
\[ = \left( \vec{a} \times \vec{b} \right) . \vec{c} - \left( \vec{a} \times \vec{b} \right) . \vec{a} - \left( \vec{a} \times \vec{c} \right) . \vec{c} + \left( \vec{a} \times \vec{c} \right) . \vec{a} + \left( \vec{b} \times \vec{c} \right) . \vec{c} - \left( \vec{b} \times \vec{c} \right) . \vec{a} \]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{a} \right] - \left[ \vec{a} \vec{c} \vec{c} \right] + \left[ \vec{a} \vec{c} \vec{a} \right] + \left[ \vec{b} \vec{c} \vec{c} \right] - \left[ \vec{b} \vec{c} \vec{a} \right]\]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - 0 - 0 + 0 + 0 - \left[ \vec{a} \vec{b} \vec{c} \right] \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left[ \vec{b} \vec{c} \vec{a} \right] = \left[ \vec{c} \vec{a} \vec{b} \right] \right)\]
\[ = 0 \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \]
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