A) \[(\overrightarrow{a}\times \overrightarrow{c})\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{0}\]
B) \[(\overrightarrow{a}\times \overrightarrow{c})\cdot (\overrightarrow{b}\times \overrightarrow{c})=0\]
C) \[(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=\overrightarrow{0}\]
D) \[(\overrightarrow{a}\times \overrightarrow{b})\cdot (\overrightarrow{c}\times \overrightarrow{d})=0\]
Correct Answer: C
Solution :
Since, \[\vec{a}\] and \[\vec{b}\] are coplanar, therefore \[\vec{a}\times \vec{b}\] is a vector perpendicular to the plane containing \[\vec{a}\] and \[\vec{b}\]. Similarly, \[\vec{c}\times \vec{d}\] is a vector perpendicular to the plane containing\[\vec{c}\]and\[\vec{d}\]. Thus, the two planes will be parallel if their normal, i.e.,\[(\vec{a}\times \vec{b})\]and\[(\vec{c}\times \vec{d})\]are parallel. \[\Rightarrow \] \[(\vec{a}\times \vec{b})\times (\vec{c}\times \vec{d})=\vec{0}\]
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