A) \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\]
B) \[2\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\]
C) \[{{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}^{2}}\]
D) \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\]
Correct Answer: C
Solution :
\[[\mathbf{b}\times \mathbf{c}\,\,\,\mathbf{c}\times \mathbf{a}\,\,\,\mathbf{a}\times \mathbf{b}]\]\[=(\mathbf{b}\times \mathbf{c})\,.[(\mathbf{c}\times \mathbf{a})\,\times \,(\mathbf{a}\times \mathbf{b})]\] Let \[\mathbf{a}\times \mathbf{b}=\mathbf{d}\] so , \[(\mathbf{b}\times \mathbf{c})[(\mathbf{c}\times \mathbf{a})\times \mathbf{d}]=(\mathbf{b}\times \mathbf{c})[(\mathbf{d}.\,\mathbf{a})\mathbf{c}-(\mathbf{d}.\mathbf{c}).\mathbf{a}]\] \[=(\mathbf{b}\times \mathbf{c})[\mathbf{a}.(\mathbf{a}\times \mathbf{b}).\mathbf{c}-(\mathbf{a}\times \mathbf{b})\mathbf{c}.\mathbf{a}]\] \[=(\mathbf{b}\times \mathbf{c})[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\mathbf{a}=\mathbf{a}.[\mathbf{b}\times \mathbf{c}].[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] \[=[\mathbf{a}\,\mathbf{b}\,\mathbf{c}][\mathbf{a}\,\mathbf{b}\,\mathbf{c}]={{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}^{2}}\].
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