A) 0
B) \[u\,.\,(v\times w)\]
C) \[u\,.\,(w\times v)\]
D) \[3u\,.\,(v\times w)\]
Correct Answer: B
Solution :
\[(\mathbf{u}+\mathbf{v}-\mathbf{w})\,\mathbf{.}\,(\mathbf{u}\times \mathbf{v}-\mathbf{u}\times \mathbf{w}-\mathbf{v}\times \mathbf{v}+\mathbf{v}\times \mathbf{w})\] \[=(\mathbf{u}+\mathbf{v}-\mathbf{w})\,(\mathbf{u}\times \mathbf{v}-\mathbf{u}\times \mathbf{w}+\mathbf{v}\times \mathbf{w})\] \[=\frac{\mathbf{u}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{v})}{0}-\frac{\mathbf{u}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{w})}{0}+\mathbf{u}\,\mathbf{.}\,(\mathbf{v}\times \mathbf{w})+\frac{\mathbf{v}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{v})}{0}\] \[-\mathbf{v}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{w})+\frac{\mathbf{v}\,\mathbf{.}\,(\mathbf{v}\times \mathbf{w})}{0}-\mathbf{w}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{v})+\frac{\mathbf{w}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{w})}{0}\] \[-\frac{\mathbf{w}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{w})}{0}=\mathbf{u}\,\mathbf{.}\,(\mathbf{v}\times \mathbf{w})-\mathbf{v}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{w})-\mathbf{w}\,\mathbf{.}\,(\mathbf{u}\times \mathbf{v})\] \[=[\mathbf{u}\,\mathbf{v}\,\mathbf{w}]+[\mathbf{v}\,\mathbf{w}\,\mathbf{u}]-[\mathbf{w}\,\mathbf{u}\,\mathbf{v}]=\mathbf{u}\,\mathbf{.}\,(\mathbf{v}\times \mathbf{w}).\]
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