If a = 3 j + 4 k , b = 2 i + k and c , d are respectively the com… - Vidyadip

Question

If $\vec a = 3\vec j + 4\vec k$ , $\vec b = 2\vec i + \vec k$ and $\vec c$ , $\vec d$ are respectively the component of $\vec a$ parallel & perpendicular to $\vec b$ ,then $\left[ {\left( {\vec a \times \vec c} \right) \times \left( {\vec c \times \vec d} \right)\,\left( {\vec c \times \vec d} \right) \times \left( {\vec d \times \vec a} \right)\left( {\vec d \times \vec a} \right) \times \left( {\vec a \times \vec c} \right)} \right]$ equals

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Answer

d
On simplifying,

$[(\vec a \times \vec c) \times (\vec c \times \vec d)(\vec c \times \vec d) \times (\vec d \times \vec a)(\vec d \times \vec a) \times (\vec a \times \vec c)]$

$ = {\left[ {\,\vec a\,\vec c\,\vec d} \right]^4}$

But $\vec a,\vec c,\vec d$ are coplanar vectors.

$ \Rightarrow \left[ {\vec a\,\vec c\,\vec d} \right] = 0$

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