If non-coplanar vectors $\vec{a}, \vec{b}, \vec{c}$ from a parallelopiped of volume 6 cubic units, then the values of $[\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}]$ are _______________.

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$For non ‐ coplanar vectors \vec{a}, \vec{b} and \vec{c} . Given volume of parallelopiped is 6 cubic units Then [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = \{(\vec{a} + \vec{b}) \times (\vec{b} + \vec{c})\} \cdot (\vec{c} + \vec{a}) (By defination) = \{(\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c}) + (\vec{b} \times \vec{b}) + (\vec{b} \times \vec{c})\} \cdot (\vec{c} + \vec{a}) [By distributive law] = \{(\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c}) + 0 + (\vec{b} \times \vec{c})\} \cdot (\vec{c} + \vec{a}) [\vec{b} \times \vec{b} = 0] = (\vec{a} \times \vec{b}) \cdot \vec{c} + (\vec{a} \times \vec{c}) \cdot \vec{c} + (\vec{b} \times \vec{c}) \cdot \vec{c} + (\vec{a} \times \vec{b}) \cdot \vec{a} + (\vec{a} \times \vec{c}) \cdot \vec{a} + (\vec{b} \times \vec{c}) \cdot \vec{a} = (\vec{a} \times \vec{b}) \cdot \vec{c} + 0 + 0 + 0 + 0 + (\vec{b} \times \vec{c}) \cdot \vec{a} [∵ two vectors of scalar triple product is zero] = + [\vec{a} \vec{b} \vec{c}] + [\vec{a} \vec{b} \vec{c}] = \pm 2 [\vec{a} \vec{b} \vec{c}] = \pm 2 \times 6 = \pm 12 i . e [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = \pm 12$

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