Question

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

 

 

Answer 1

\[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right)\]
\[ = \left( \vec{a} \times \vec{b} \right) + \left( \vec{a} \times \vec{c} \right) + \left( \vec{b} \times \vec{c} \right) + \left( \vec{b} \times \vec{a} \right) + \left( \vec{c} \times \vec{a} \right) + \left( \vec{c} \times \vec{b} \right)\]
\[ = \left( \vec{a} \times \vec{b} \right) + \left( \vec{a} \times \vec{c} \right) + \left( \vec{b} \times \vec{c} \right) - \left( \vec{a} \times \vec{b} \right) - \left( \vec{a} \times \vec{c} \right) - \left( \vec{b} \times \vec{c} \right)\]
\[ = \vec{0}\]