If $\overline{\mathrm{a}}=2\overline{\mathrm{i}} + \overline{\mathrm{j}}-3\overline{\mathrm{k}}, \overline{\mathrm{b}} = \overline{\mathrm{i}}-2\overline{\mathrm{j}} + \overline{\mathrm{k}}, \overline{\mathrm{c}}=-\overline{\mathrm{i}} + \overline{\mathrm{j}} -4\overline{\mathrm{k}}, \overline{\mathrm{d}}= \overline{\mathrm{i}} + \overline{\mathrm{j}} + \overline{\mathrm{k}}$ then $\left|\right(\overline{\mathrm{a}}\times \overline{\mathrm{b}} )\times (\overline{\mathrm{c}}\times \overline{\mathrm{d}}\left)\right|=$

$\begin{array}{rcl}\overline{\mathrm{a}}\times \overline{\mathrm{b}}& =& \left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& 3\\ 1& -2& 1\end{array}\right|=\mathrm{i}(1-6)-\mathrm{j}(2+3)+\mathrm{k}(-4-1)\\ & =& 5(\mathrm{i}-\mathrm{i}-\mathrm{k}) \\ \overline{\mathrm{c}} \times \overline{\mathrm{d}}& =& \left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -5& -5& -5\\ 5& -3& -2\end{array}\right|\\ & =& \mathrm{i}(10-15)-\mathrm{j}(10+25)+\mathrm{k}(15+25)\\ & =& -5\mathrm{i}-35\mathrm{j}+40\mathrm{k}\\ & =& -5(\mathrm{i}+7\mathrm{j}-8\mathrm{k})\\ \left|\right(\overline{\mathrm{a}}\times \overline{\mathrm{b}})\times (\overline{\mathrm{c}}\times \overline{\mathrm{d}}\left)\right|& =& 5\sqrt{1+49+64}=5\sqrt{114}\end{array}$