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If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be any three non-coplanar vectors, then $[\mathbf{a}+\mathbf{b} \mathbf{b}+\mathbf{c} \mathbf{c}+\mathbf{a}]=$
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The correct answer is:
$2|\mathbf{a b} \mathbf{c}|$
$\begin{aligned} & {[\mathbf{a}+\mathbf{b} \mathbf{b}+\mathbf{c} \mathbf{c}+\mathbf{a}]=(\mathbf{a}+\mathbf{b}) .\{(\mathbf{b}+\mathbf{c}) \times(\mathbf{c}+\mathbf{a})\}} \\ & =(\mathbf{a}+\mathbf{b}) .(\mathbf{b} \times \mathbf{c}+\mathbf{b} \times \mathbf{a}+\mathbf{c} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}) \\ & =(\mathbf{a}+\mathbf{b}) .(\mathbf{b} \times \mathbf{c}+\mathbf{b} \times \mathbf{a}+\mathbf{c} \times \mathbf{a}), \quad[\mathbf{c} \times \mathbf{c}=0\} \\ & =\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}+\mathbf{a} \cdot \mathbf{b} \times \mathbf{a}+\mathbf{a} \cdot \mathbf{c} \times \mathbf{a}+\mathbf{b} . \mathbf{b} \times \mathbf{c}+\mathbf{b} \cdot \mathbf{b} \times \mathbf{a}+\mathbf{b} . \mathbf{c} \times \mathbf{a} \\ & =[\mathbf{a} \mathbf{b} \mathbf{c}]+[\mathbf{b} \mathbf{c} \mathbf{a}]=2[\mathbf{a} \mathbf{b} \mathbf{c}]\end{aligned}$
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