<p>If <span class="math-tex">\(\rm \vec{a},\vec{b},\vec{c}\)</span> are three non-coplanar vectors, then</p> <p><span class="math-tex">\(\rm (\vec{a}+\vec{b}+\vec{c}) \cdot[(\vec{a}+\vec{b}) \times( \vec{a}+\vec{c})]=\)</span></p>
\[ (\vec a + \vec b + \vec c) \cdot \left[ \left( \vec a + \vec b \right) \times \left( \vec a + \vec c \right) \right] \] This can be expanded as: \[ (\vec a + \vec b + \vec c) \cdot \left[ \vec a \times \vec a + \vec a \times \vec c + \vec b \times \vec a + \vec b \times \vec c \right] \] This simplifies to: \[ \vec a \cdot (\vec b \times \vec c) + \vec b \cdot (\vec a \times \vec c) + \vec c \cdot (\vec b \times \vec a) \] And further: \[ \left[ \vec a \, \vec b \, \vec c \right] + \left[ \vec b \, \vec a \, \vec c \right] + \left[ \vec c \, \vec b \, \vec a \right] = - \left[ \vec a \, \vec b \, \vec c \right] \]
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