Let $\overline{\mathrm{a}}$ be a non zero vector. If $\overline{\mathrm{x}}=\overline{\mathrm{i}}\times (\overline{\mathrm{a}}\times \overline{\mathrm{i}} ), \overline{\mathrm{y}}=\overline{\mathrm{j}}\times (\overline{\mathrm{a}}\times \overline{\mathrm{j}} )-\overline{\mathrm{a}} \mathrm{and} \overline{\mathrm{z}}=\overline{\mathrm{k}}\times (\overline{\mathrm{a}}\times \overline{\mathrm{k}})-\overline{\mathrm{a}}$, then $[\overline{\mathrm{x}} \overline{\mathrm{y}} \overline{\mathrm{z}} ]=$

$\begin{array}{rcl}\overline{\mathrm{a}}& =& {\mathrm{a}}_1\mathrm{i}+{\mathrm{b}}_1\mathrm{j}+{\mathrm{c}}_1\mathrm{k}\\ \overline{\mathrm{x}}& =& \overline{\mathrm{a}}-(\overline{\mathrm{a}}·\mathrm{i}) \mathrm{i} \overline{\mathrm{b}}=\overline{\mathrm{a}}-(\overline{\mathrm{a}}·\mathrm{j}) \mathrm{j}-\overline{\mathrm{a}}\\ & =& -(\overline{\mathrm{a}}·\mathrm{j}) \mathrm{j}\\ \overline{\mathrm{z}}& =& \overline{\mathrm{a}}-(\overline{\mathrm{a}}·\mathrm{k}) \mathrm{k}-\overline{\mathrm{a}} =-(\overline{\mathrm{a}}·\mathrm{k})\\ \overline{\mathrm{x}}& =& {\mathrm{b}}_1\mathrm{j}+{\mathrm{c}}_1\mathrm{k} \overline{\mathrm{b}}=-{\mathrm{b}}_1\mathrm{j} \overline{\mathrm{z}}=-{\mathrm{c}}_1\mathrm{k}\\ [\overline{\mathrm{x}} \overline{\mathrm{y}} \overline{\mathrm{z}}]& =& \left|\begin{array}{ccc}0& {\mathrm{b}}_1& {\mathrm{c}}_1\\ 0& -{\mathrm{b}}_1& 0\\ 0& 0& -{\mathrm{c}}_1\end{array}\right|=0\end{array}$