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

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