A) \[|\mathbf{a}\,\mathbf{b}\,\mathbf{c}|\]
B) 2\[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\]
C) \[{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\]
D) \[2\,{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\]
Correct Answer: B
Solution :
\[[\mathbf{a}+\mathbf{b}\,\,\mathbf{b}+\mathbf{c}\,\,\mathbf{c}+\mathbf{a}]=(\mathbf{a}+\mathbf{b}).\{(\mathbf{b}+\mathbf{c})\times (\mathbf{c}+\mathbf{a})\}\] \[=(\mathbf{a}+\mathbf{b}).(\mathbf{b}\times \mathbf{c}+\mathbf{b}\times \mathbf{a}+\mathbf{c}\times \mathbf{c}+\mathbf{c}\times \mathbf{a})\] \[=(\mathbf{a}+\mathbf{b}).(\mathbf{b}\times \mathbf{c}+\mathbf{b}\times \mathbf{a}+\mathbf{c}\times \mathbf{a})\], \[\left\{ \because \,\mathbf{c}\times \mathbf{c}=0 \right\}\] \[=\mathbf{a}.\mathbf{b}\times \mathbf{c}+\mathbf{a}.\mathbf{b}\times \mathbf{a}+\mathbf{a}.\mathbf{c}\times \mathbf{a}+\mathbf{b}.\mathbf{b}\times \mathbf{c}\]\[+\mathbf{b}.\mathbf{b}\times \mathbf{a}+\mathbf{b}.\mathbf{c}\times \mathbf{a}\] \[=[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]+[\mathbf{b}\,\mathbf{c}\,\mathbf{a}]=2[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\].
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