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Question
For any three vectors \[\vec{a,} \vec{b} \text{ and } \vec{c}\] write the value of \[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right) .\]
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Solution
\[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right)\]
\[ = \left( \vec{a} \times \vec{b} \right) + \left( \vec{a} \times \vec{c} \right) + \left( \vec{b} \times \vec{c} \right) + \left( \vec{b} \times \vec{a} \right) + \left( \vec{c} \times \vec{a} \right) + \left( \vec{c} \times \vec{b} \right)\]
\[ = \left( \vec{a} \times \vec{b} \right) + \left( \vec{a} \times \vec{c} \right) + \left( \vec{b} \times \vec{c} \right) - \left( \vec{a} \times \vec{b} \right) - \left( \vec{a} \times \vec{c} \right) - \left( \vec{b} \times \vec{c} \right)\]
\[ = \vec{0}\]
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