For any three vectors a.b.c. write the value of the following: a cross (b+c) + b cross (c+a) + c cross (a+b).

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Solution

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

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