If four distinct points with position vectors  $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$ are coplanar, then $\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]$ is equal to

$$\begin{gathered} (\overline{\mathrm{a}}-\overline{\mathrm{b}}).(\overline{\mathrm{a}}-\overline{\mathrm{c}})\times (\overline{\mathrm{a}}-\overline{\mathrm{d}})=0\Rightarrow (\overline{\mathrm{a}}-\overline{\mathrm{b}}).(-\overline{\mathrm{a}}\times \overline{\mathrm{d}}-\overline{\mathrm{c}}\times \overline{\mathrm{a}}+\overline{\mathrm{c}}\times \overline{\mathrm{d}})=0 \\ \left[\overline{\mathrm{a}}\overline{\mathrm{c}}\overline{\mathrm{d}}\right]+\left[\overline{\mathrm{b}}\overline{\mathrm{a}}\overline{\mathrm{d}}\right]+\left[\overline{\mathrm{b}}\overline{\mathrm{c}}\overline{\mathrm{a}}\right]-\left[\overline{\mathrm{b}}\overline{\mathrm{c}}\overline{\mathrm{d}}\right]=0 \\ \left[\overline{\mathrm{a}}\overline{\mathrm{b}}\overline{\mathrm{c}}\right]=\left[\overline{\mathrm{d}}\overline{\mathrm{c}}\overline{\mathrm{a}}\right]+\left[\overline{\mathrm{b}}\overline{\mathrm{d}}\overline{\mathrm{a}}\right]+\left[\overline{\mathrm{c}}\overline{\mathrm{d}}\overline{\mathrm{b}}\right] \end{gathered}$$