Let $\overline{\mathrm{a}}=2\overline{\mathrm{i}}+\overline{\mathrm{j}}-2\overline{\mathrm{k}}, \overline{\mathrm{b}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}$. If $\overline{\mathrm{c}}$ is a vector such that $\overline{\mathrm{a}}.\overline{\mathrm{c}}=\left|\overline{\mathrm{c}}\right|, |\overline{\mathrm{c}}-\overline{\mathrm{a}}|=2\sqrt{2}$ and the angle between $(\overline{\mathrm{a}}\times \overline{\mathrm{b}})$ and $\overline{\mathrm{c}}$ is ${30}^0$, then $\left|\right(\overline{\mathrm{a}}\times \overline{\mathrm{b}} )\times \overline{\mathrm{c}}|=$

$\begin{array}{rcl}\overline{\mathrm{a}}\times \overline{\mathrm{b}}& =& \left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& -2\\ 1& 1& 0\end{array}\right|=\mathrm{i}(0+2)-\mathrm{j}(0+2)+\mathrm{k}(2-1)\\ & =& 2\mathrm{i}-2\mathrm{j}+\mathrm{k}\\ |\overline{\mathrm{a}}\times \overline{\mathrm{b}}|& =& \sqrt{4+4+1}=3\\ |\overline{\mathrm{c}}-\overline{\mathrm{a}}{|}^2& =& 8 \Rightarrow |\overline{\mathrm{c}}{|}^2+9-2|\overline{\mathrm{c}}|=8\\ & \Rightarrow & |\overline{\mathrm{c}}{|}^2-2|\overline{\mathrm{c}}|+1=0 \Rightarrow {\left(\right|\overline{\mathrm{c}}|-1)}^2=0 \Rightarrow |\overline{\mathrm{c}}|=1\\ \left|\right(\overline{\mathrm{a}}\times \overline{\mathrm{b}})\times \overline{\mathrm{c}}|& =& |\overline{\mathrm{a}}\times \overline{\mathrm{b}}| \left|\overline{\mathrm{c}}\right| \sin {30}^0\\ & =& 3\times 1\times \frac{1}{2}=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}$