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