If $\to a, \to b, \to c$ are three non coplanar vectors represented by concurrent edges of a parallelepiped of volume $4$ then
$(\to a + \to b) . (\to b \times \to c) + (\to b + \to c) . (\to c \times \to a) + (\to c + \to a) . (\to a \times \to b)$ is equal to

12

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4

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\pm 12

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0

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Solution

The correct option is C $\pm 12$
$[\to a \to b \to c] = \pm 4$ from the definition of volume of a parallelopipe

Thus, $(\to a + \to b) . (\to b \times \to c) + (\to b + \to c) \cdot (\to c \times \to a) + (\to c + \to a) \cdot (\to a \times \to b) =$

$[\to a \to b \to c] + 0 + [\to b \to c \to a] + 0 + [\to c \to a \to b] + 0$

$= 3 [\to a \to b \to c]$

$= \pm 12$

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