If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are non-coplanar, nonzero vectors and $\overline{\mathrm{r}}$ is any vector in space then $[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{r}}]\overline{\mathrm{c}}+[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{r}}]\overline{\mathrm{a}}+[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{r}}]\overline{\mathrm{b}}=$

Let  $\overline{\mathrm{r}}=\mathrm{x} \overline{\mathrm{a}}+\mathrm{y} \overline{\mathrm{b}}+\mathrm{z} \overline{\mathrm{c}}$

$\overline{\mathrm{b}}\times \overline{\mathrm{c}}·\overline{\mathrm{r}}=\mathrm{x}[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{a}}] \Rightarrow \mathrm{x}=\frac{[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{r}}]}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}$

Similarly  $\begin{array}{rcl}\mathrm{y}& =& \frac{[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{r}}]}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]} \mathrm{z}=\frac{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{r}}]}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\end{array}$

    $\begin{array}{rcl}\overline{\mathrm{r}}& =& \frac{[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{r}}] \overline{\mathrm{a}}+[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{r}}] \overline{\mathbf{b}}+[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{r}}] \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\\ [\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}] \overline{\mathrm{r}}& =& [\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{r}}] \overline{\mathrm{a}}+[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{r}}] \overline{\mathrm{b}}+[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{r}} ] \overline{\mathrm{c}}\end{array}$