A) \[|\,\mathbf{a}\,\times \,\mathbf{c}{{|}^{2}}\]
B) \[|\mathbf{a}\times \mathbf{d}{{|}^{2}}\]
C) \[|\mathbf{b}\times \mathbf{c}{{|}^{2}}\]
D) 0
Correct Answer: D
Solution :
\[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=[\mathbf{a}\,\mathbf{b}\,\mathbf{d}]\mathbf{c}-[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\mathbf{d}\] \[\because \,\,\,\mathbf{a},\,\mathbf{b},\,\mathbf{c},\,\mathbf{d}\] are coplanar vectors \[\therefore \,\,\,[\mathbf{a}\,\mathbf{b}\,\mathbf{d}]=[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]=0.\] So, \[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=\mathbf{0}\].
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