A vector of magnitude $\sqrt{2}$ coplanar with the vectors $\overrightarrow{\mathrm{a}}=\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\text{ and }\overrightarrow{\mathrm{b}}=\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}+\widehat{\mathrm{k}}$  and perpendicular to the vector $\overrightarrow{\mathrm{c}}=\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+\widehat{\mathrm{k}},$ is 

A vector coplanar with $\overrightarrow{\mathrm{a}}\text{ and }\overrightarrow{\mathrm{b}}$ and perpendicular  to $\overrightarrow{\mathrm{c}}\text{ is }\mathrm{\lambda }\left(\right(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}})$

$\begin{array}{l}\mathrm{but} \mathrm{\lambda }\left(\right(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}})=\mathrm{\lambda }\left[\right(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}\left)\overrightarrow{\mathrm{a}}\right]\\ =\mathrm{\lambda }[4\overrightarrow{\mathrm{b}}-4\overrightarrow{\mathrm{a}}]\\ =4\mathrm{\lambda }[\widehat{\mathrm{j}}-\widehat{\mathrm{k}}]\\ \mathrm{now} 4\left|\mathrm{\lambda }\right|\sqrt{2}=\sqrt{2}\text{ (Given) }\Rightarrow \mathrm{\lambda }=\pm \frac{1}{4}\end{array}$

Hence, the required vector is $\widehat{j}-\widehat{k}\text{ or }-\widehat{j}+\widehat{k}$