A) \[0\]
B) \[1\]
C) \[2\]
D) \[3\]
Correct Answer: C
Solution :
\[[\vec{a}+\vec{b}\,\,\,\,\,\vec{b}+\vec{c}\,\,\,\,\,\vec{c}+\vec{a}]\] \[=(\vec{a}+\vec{b}).(\vec{b}+\vec{c})\times (\vec{c}+\vec{a})\] \[=(\vec{a}+\vec{b}).(\vec{b}\times \vec{c}+\vec{b}\times \vec{a}+\vec{c}\times \vec{c}+\vec{c}\times \vec{a})\] \[=(\vec{a}+\vec{b}).(\vec{b}\times \vec{c}+\vec{b}\times \vec{a}+\vec{c}\times \vec{a})\] \[=\vec{a}.(\vec{b}\times \vec{c})+\vec{a}.(\vec{b}\times \vec{a})+\vec{a}.(\vec{c}\times \vec{a})\] \[+\vec{b}.(\vec{b}\times \vec{c})+\vec{b}.(\vec{b}\times \vec{a})+\vec{b}.(\vec{c}\times \vec{a})\] \[=2[\vec{a}\,\,\vec{b}\,\,\vec{c}]\] \[\therefore \] \[k=2\]
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