A) \[[\overrightarrow{A}\,\overrightarrow{B}\,\overrightarrow{C}]\]
B) \[[\overrightarrow{B}\,\overrightarrow{A}\,\overrightarrow{C}]\]
C) 0
D) 1
Correct Answer: C
Solution :
\[\overrightarrow{A}.\{(\overrightarrow{B}+\overrightarrow{C})\times (\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C})\}\] \[=\overrightarrow{A}.\{\overrightarrow{B}\times \overrightarrow{A}+\overrightarrow{0}+\overrightarrow{B}\times \overrightarrow{C}+\overrightarrow{C}\times \overrightarrow{A}+\overrightarrow{C}\times \overrightarrow{B}+0\}\] \[=[\overrightarrow{A}\,\overrightarrow{B}\overrightarrow{A}]+[\overrightarrow{A}\overrightarrow{B}\overrightarrow{C}]+[\overrightarrow{A}\,\overrightarrow{C}\overrightarrow{A}]-[\overrightarrow{A}\,\overrightarrow{B}\,\overrightarrow{C}]\] \[=0\]
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