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Test for Coplanarity

If a⃗,b⃗ and ...

Question

If $\to a, \to b$ and $\to c$ are three non – coplanar vectors, then $(\to a + \to b + \to c) . ((\to a + \to b) \times (\to a + \to c))$ equals

A

\to 0

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B

[\to a \to b \to c]

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C

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

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D

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

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Solution

The correct option is D $- [\to a \to b \to c]$
We have, $(\to a + \to b) \times (\to a + \to c) = \to a \times \to c + \to b \times \to a + \to b \times \to c$
$= - \to a \times b + \to b \times \to c - \to c \times \to a ∴ (\to a + \to b + \to c) . {(\to a + \to b) \times (\to a + \to c)}$
$= (\to a + \to b + \to c) . (- \to a \times \to b + \to b \times \to c - \to c \times \to a)$
$= [\to a \to a \to b] + [\to a \to b \to c] - [\to a \to c \to a] - [\to b \to a \to b] + [\to b \to b \to c] - [\to b \to c \to a] - [\to c \to a \to b] + [\to c \to b \to c] - [\to c \to c \to a] = [\to a \to b \to c] - [\to b \to c \to a] - [\to c \to a \to b] = - [\to a \to b \to c]$

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