Question

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

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

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

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

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

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

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

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

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

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

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

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