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Vector Addition

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Question

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $[\vec{a}, \vec{b}, \vec{c}] = 10, then [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}]$ = _______________.

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Solution

$for vectors \vec{a}, \vec{b} and \vec{c,} given [\vec{a} \vec{b} \vec{c}] = 10 then [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = \{(\vec{a} + \vec{b}) \times (\vec{b} + \vec{c})\} \cdot (\vec{c} + \vec{a}) (defination) = (\vec{a} \times \vec{b} + \vec{a} \times \vec{c} + \vec{b} \times \vec{b} + \vec{b} \times \vec{c}) \cdot (\vec{c} + \vec{a}) [By distribution law] = (\vec{a} \times \vec{b} + \vec{a} \times \vec{c} + \vec{b} \times \vec{c}) \cdot (\vec{c} + \vec{a}) [∵ \vec{b} \times \vec{b} = 0] = (\vec{a} \times \vec{b}) \cdot \vec{c} + (\vec{a} \times \vec{c}) \cdot \vec{c} + (\vec{b} \times \vec{c}) \cdot \vec{c} + (\vec{a} \times \vec{b}) \cdot \vec{a} + (\vec{a} \times \vec{c}) \cdot \vec{a} + (\vec{b} \times \vec{c}) \cdot \vec{a}$
$= [\vec{a} \vec{b} \vec{c}] + [\vec{a} \vec{c} \vec{c}] + [\vec{b} \vec{c} \vec{c}] + [\vec{a} \vec{b} \vec{a}] + [\vec{a} \vec{c} \vec{a}] + [\vec{b} \vec{c} \vec{a}] [Since scalar triple product when any two vectors are same is zero .] = [\vec{a} \vec{b} \vec{c}] + 0 + 0 + 0 + 0 + [\vec{a} \vec{b} \vec{c}] = 2 [\vec{a} \vec{b} \vec{c}] = 2 \times 10 = 20 i . e [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = 20$

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