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