Question

If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar vectors, then \[\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]\] equals

Options

• 0

• \[\left[ \vec{a} \vec{b} \vec{c} \right]\]

• \[2\left[ \vec{a} \vec{b} \vec{c} \right]\]

• \[- \left[ \vec{a} \vec{b} \vec{c} \right]\]

Answer 1

\[ - \left[ \vec{a} \vec{b} \vec{c} \right]\]

We have

\[\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]\]

\[ = \left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \vec{a} + \left( \vec{a} + \vec{b} \right) \times \vec{c} \right] \left(\text {  By definition of cross poduct }\right)\]

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

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

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

\[ = \left[ \vec{a} \vec{b} \vec{a} \right] + \left[ \vec{a} \vec{a} \vec{c} \right] + \left[ \vec{a} \vec{b} \vec{c} \right] + \left[ \vec{b} \vec{b} \vec{a} \right] + \left[ \vec{b} \vec{a} \vec{c} \right] + \left[ \vec{b} \vec{b} \vec{c} \right] + \left[ \vec{c} \vec{b} \vec{a} \right] + \left[ \vec{c} \vec{a} \vec{c} \right] + \left[ \vec{c} \vec{b} \vec{c} \right]\]

\[ = 0 + 0 + \left[ \vec{a} \vec{b} \vec{c} \right] + 0 + \left[ \vec{b} \vec{a} \vec{c} \right] + 0 + \left[ \vec{c} \vec{b} \vec{a} \right] + 0 + 0\]

\[ = \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{c} \right] \left( \because \left[ \vec{b} \vec{a} \vec{c} \right] = - \left[ \vec{c} \vec{a} \vec{b} \right], \left[ \vec{b} \vec{a} \vec{c} \right] = - \left[ \vec{a} \vec{b} \vec{c} \right] \right)\]

\[ = - \left[ \vec{a} \vec{b} \vec{c} \right]\]