If $\to a, \to b, \to c$ be any three non - coplanar vectors, then calculate the value of $[\to a + \to b \to b + \to c \to c + \to a]$ is

2 [\to a + \to b + \to c]

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

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2 \sqrt{[\to a + \to b + \to c]}

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

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Solution

The correct option is B $2 [\to a \to b \to c]$
$[\to a + \to b \to b + \to c \to c + \to a] = (\to a + \to b) . {(\to b + \to c) \times (\to c + \to a)}$
$= (\to a + \to b) . (\to b \times \to c + \to b \times \to a + \to c \times \to c + \to c \times \to a)$
$= (\to a + \to b) . (\to b \times \to c + \to b \times \to a + \to c \times \to a)$
$= \to a (\to b \times \to c) + \to a (\to b \times \to a) + \to a . (\to c \times \to a) + \to b . (\to b \times \to c) + \to b (\to b \times \to a) + \to b . (\to c \times \to a)$
$= [\to a \to b \to c] + [\to b \to c \to a]$
$= 2 [\to a \to b \to c]$

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