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The correct expression in the following

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$\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = 0$

$\vec{a} . (\vec{b} . \vec{c}) + \vec{b} . (\vec{c} . \vec{a}) + \vec{c} . (\vec{a} . \vec{b}) = 0$

$\vec{a} . (\vec{b} \times \vec{c}) + \vec{b} . (\vec{c} \times \vec{a}) + \vec{c} . (\vec{a} \times \vec{b}) = 0$

$\vec{a} . (\vec{b} + \vec{c}) + \vec{b} . (\vec{c} + \vec{a}) + \vec{c} . (\vec{a} + \vec{b}) = 0$

answer is A.

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Detailed Solution

$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} . \vec{c}) \vec{b} - (\vec{a} . \vec{b}) \vec{c}$

$\vec{b} \times (\vec{c} \times \vec{a}) = (\vec{b} . \vec{a}) \vec{c} - (\vec{b} . \vec{c}) \vec{a}$

$\vec{c} \times (\vec{a} \times \vec{b}) = (\vec{c} . \vec{b}) \vec{a} - (\vec{c} . \vec{a}) \vec{b}$

$\Rightarrow$ $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = \vec{0}$

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