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Question

If $$a,c,d$$ are non-coplanar vectors and $$d.\left\{ a\times \left[ b\times \left( c\times d \right)  \right]  \right\} $$ is equal to


A
(b.d)[acd]
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B
(a.d)[acd]
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C
(c.d)[acd]
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D
None of these
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Solution

The correct option is D $$\left( b.d \right) \left[ a\quad c\quad d \right] $$
We have $$d.\left\{ a\times \left[ b\times \left( c\times d \right)  \right]  \right\} $$
$$=d.\left( a\times \left( \left( b.d \right) c-\left( b.c \right) d \right)  \right) \\ =d.\left( \left( b.d \right) \left( a\times c \right) -\left( b.c \right) \left( a\times d \right)  \right) \\ =\left( b.d \right) d.\left( a\times c \right) -\left( b.c \right) d.\left( a\times d \right) \\ =\left( b.d \right) \left[ a\quad c\quad d \right] \quad \quad \quad \left[ \because d.\left( a\times d \right) =0 \right] $$

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