If $\to a, \to b, \to c$ are three non-coplanar, non -zero vectors , then the value of $(\to a . \to a) \to b \times \to c + (\to a . \to b) \to c \times \to a + (\to a . \to c) \to a \times \to b$

[\to b \to c \to a] \to a

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[\to c \to a \to b] \to b

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[\to a \to b \to c] \to c

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[\to a \to c \to b] \to a

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Solution

The correct option is A $[\to b \to c \to a] \to a$
Let $\to A = (\to a . \to a) \to b \times \to c + (\to a . \to b) \to c \times \to a + (\to a \to c) \to a \times \to b$
$\to A . \to a = (\to a . \to a) (\to a . \to b \times \to c) + 0 = (\to a . \to a) [\to a \to b \to c]$
$\to A . \to b = (\to a . \to b) (\to b . \to c \times \to a) = (\to a \to b) [\to a \to b \to c]$ $[∵ (\to a . \to a \to b) = 0]$
$\to A . \to c = (\to a . \to c) (\to c . \to a \times \to b) = (\to a . \to c) [\to a \to b \to c]$ $[\to b \to a \to b] = 0$
Adding , $\to A . \to a + \to A . \to b + \to A . \to c = (\to a \to b \to c) (\to a . \to a + \to a . \to b + \to a . \to c)$
$\Rightarrow \to A (\to a + \to b + \to c) = [\to a \to b \to c] \to a (\to a + \to b + \to c)$
$(\to A - [\to a \to b \to c] \to a) (\to a + \to b + \to c) = 0$
As $\to a, \to b, \to c$ are non-coplanar $\Rightarrow \to A = [\to a \to b \to c] \to a$
As $| \to a \to b \to c | = [\to b \to c \to a]$
$\Rightarrow \to A = [\to b \to c \to a]$
$∴$ Option $A$ is correct

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