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Collinear Vectors

If a, b and c...

Question

If $\to a, \to b$ and $\to c$ are any three non-zero vectors in space, then the value of $(\to c + \to b) \times (\to c + \to a) \cdot (\to c + \to b + \to a)$ is :

A

[\to a \to b \to c]

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B

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

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C

0

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D

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

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Solution

The correct option is A $[\to a \to b \to c]$
Let $E = (\to c + \to b) \times (\to c + \to a) \cdot (\to c + \to b + \to a)$
$\Rightarrow E = (\to c \times \to c + \to c \times \to a + \to b \times \to c + \to b \times \to a) \cdot (\to c + \to b + \to a)$
Since $\to c \times \to c = \to 0$ ,
$\Rightarrow E = (\to c \times \to a + \to b \times \to c + \to b \times \to a) \cdot (\to c + \to b + \to a)$
On multiplying (taking dot product) term by term, we have
$\Rightarrow E = \to b \times \to a \cdot \to c + \to c \times \to a \cdot \to b + \to b \times \to c \cdot \to a$
$\Rightarrow E = - [\to a \to b \to c] + [\to a \to b \to c] + [\to a \to b \to c]$
$\Rightarrow E = [\to a \to b \to c]$

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