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