A) \[2|\vec{a}\vec{b}\vec{c}|\]
B) \[|\vec{a}\vec{b}\vec{c}|\]
C) \[3|\vec{a}\vec{b}\vec{c}|\]
D) 0
Correct Answer: D
Solution :
Key Idea: Since, \[\vec{a},\vec{b}\]and \[\vec{c}\]are coplanar vectors, then the scalar triple product will be zero. \[\therefore \] \[[\vec{a}+\vec{b}\vec{b}+\vec{c}\vec{c}+\vec{a}]\] \[=(\vec{a}+\vec{b}).[(\vec{b}+\vec{c})\times (\vec{c}\times \vec{a})]\] \[=(\vec{a}+\vec{b}).[\vec{b}\times \vec{c}+\vec{b}\times \vec{a}+\vec{c}\times \vec{c}+\vec{c}\times \vec{a}]\] \[=[\vec{a}\vec{b}\vec{c}]+[\vec{a}\vec{b}\vec{a}]+[\vec{a}\vec{c}\vec{a}]+[\vec{b}\vec{b}\vec{c}]\] \[+[\vec{b}\vec{b}\vec{a}]+[\vec{b}\vec{c}\vec{a}]\] \[=2[\vec{a}\vec{b}\vec{c}]=0\] [\[\because \,\vec{a},\vec{b}\]and \[\vec{c}\]]
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