A) \[2[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]\]
B) 0
C) \[[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]\]
D) None of these
Correct Answer: B
Solution :
\[\overrightarrow{a}.\{(\overrightarrow{b}+\overrightarrow{c})\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\}\] \[=\overrightarrow{a}.\{\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}\] \[+\overrightarrow{c}\times \overrightarrow{b}+\overrightarrow{c}\times \overrightarrow{c}\}\] \[=\overrightarrow{a}.\{\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{c}\times \overrightarrow{b}\}\] \[=\overrightarrow{a}.\{\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}-\overrightarrow{b}\times \overrightarrow{c}\}\] \[=\overrightarrow{a}.\{\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{c}\times \overrightarrow{a}\}\] \[=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{a})+\overrightarrow{a}(\overrightarrow{c}\times \overrightarrow{a}\}\] \[=[\overrightarrow{a}\,\overrightarrow{b}\,\overrightarrow{c}]+[\overrightarrow{a}\,\overrightarrow{c}\,\overrightarrow{a}]=0\]
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