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

Solution

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

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