A) [a b c]
B) 2[a b c]
C) 3[a b c]
D) 0
Correct Answer: D
Solution :
\[\mathbf{a}\,\mathbf{.}\,[(\mathbf{b}+\mathbf{c})\times (\mathbf{a}+\mathbf{b}+\mathbf{c})]\] \[=\mathbf{a}\,\mathbf{.}\,(\mathbf{b}\times \mathbf{a}+\mathbf{b}\times \mathbf{b}+\mathbf{b}\times \mathbf{c})+\mathbf{a}\,\mathbf{.}\,(\mathbf{c}\times \mathbf{a}+\mathbf{c}\times \mathbf{b}+\mathbf{c}\times \mathbf{c})\] \[=[\mathbf{a}\,\mathbf{b}\,\mathbf{a}]+[\mathbf{a}\,\mathbf{b}\,\mathbf{b}]+[\mathbf{a}\text{ }\mathbf{b}\text{ }\mathbf{c}]+[\mathbf{a}\,\mathbf{c}\,\mathbf{a}]+[\mathbf{a}\text{ }\mathbf{c}\text{ }\mathbf{b}]+[\mathbf{a}\text{ }\mathbf{c}\text{ }\mathbf{c}]\] \[0+0+[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]+0-[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]+0=0.\]
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