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

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