$\text{ Let }\overrightarrow{a}=2\widehat{i}+\widehat{j}-2\widehat{k}\text{ and }b=\widehat{i}+\widehat{j}\text{. If }c\text{ is a vector such that }\overrightarrow{a}\cdot \overrightarrow{c}=\left|\overrightarrow{c}\right|,|\overrightarrow{c}-\overrightarrow{a}|=2\sqrt{2}\text{ and the angle between }$

$\overrightarrow{a}\times \overrightarrow{b}\text{ and }\overrightarrow{c}\text{ is }{30}^{\circ }\text{, then }\left|\right(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}|\text{ is equal to }$

$\begin{array}{r}\left|\right(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|\left|\overrightarrow{\mathrm{c}}\right|\sin {30}^{\circ }\\ =\frac{1}{2}|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|\left|\overrightarrow{\mathrm{c}}\right|----\left(1\right)\end{array}$

$\text{ We have, }\overrightarrow{a}=2\widehat{i}+\widehat{j}-2\widehat{k}\text{ and }\overrightarrow{b}=\widehat{i}+\widehat{j}$

$\begin{array}{r}\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=2\widehat{i}-2\widehat{j}+\widehat{k}\\ \text{ or }|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{9}=3\end{array}$

$\text{ Also given }|\overrightarrow{c}-\overrightarrow{a}|=2\sqrt{2}$

$\text{ or }|\overrightarrow{c}-\overrightarrow{a}{|}^2=8$

$\text{ or }|\overrightarrow{c}{|}^2+|\overrightarrow{a}{|}^2-2\overrightarrow{a}\cdot \overrightarrow{c}=8$

$\text{ Given }\left|\overrightarrow{a}\right|=3\text{ and }\overrightarrow{a}\cdot \overrightarrow{c}=\left|\overrightarrow{c}\right|,\text{ using these we get }$

$|\overrightarrow{c}{|}^2-2|\overrightarrow{c}|+1=0$

$\begin{array}{l}\mathrm{or} \left(\right|\overrightarrow{\mathrm{c}}|-1{)}^2=0\\ \mathrm{or} \left|\overrightarrow{\mathrm{c}}\right|=1\end{array}$

$\text{ Substituting values of }|\overrightarrow{a}\times \overrightarrow{b}|\text{ and }\left|\overrightarrow{c}\right|\text{ in (i), we get }$

$\left|\right(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}|=\frac{1}{2}\times 3\times 1=\frac{3}{2}$