If $\overrightarrow{\mathrm{P}}= \hat{i} + \hat{j} + 2\hat{k}$  and $\overrightarrow{\mathrm{Q}}= 3\hat{i} - 2\hat{j} + \hat{k}$ , the unit vector perpendicular to both $\overrightarrow{\mathrm{P}} \mathrm{and} \overrightarrow{\mathrm{Q}}$.

$\overline{\mathrm{P}}=\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\overline{\mathrm{Q}}=3\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+\widehat{\mathrm{k}}$
Unit vector $\perp \mathrm{r}\text{ to }\overline{\mathrm{P}}\mathrm{\&}\overline{\mathrm{Q}}\text{ is }\frac{\overline{\mathrm{P}}\times \overline{\mathrm{Q}}}{|\overline{\mathrm{P}}\times \overline{\mathrm{Q}}|}$
$\begin{array}{l}\overline{\mathrm{P}}\times \overline{\mathrm{Q}}=\left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 1& 2\\ 3& -2& 1\end{array}\right|\\ =\mathrm{i}(1+4)-\mathrm{j}(1-6)+\mathrm{k}(-2-3)=5\widehat{\mathrm{i}}+5\widehat{\mathrm{j}}-5\widehat{\mathrm{k}}\\ |\overline{\mathrm{P}}\times \overline{\mathrm{Q}}|=\sqrt{5^2+5^2+(-5{)}^2}=5\sqrt{3}\\ \therefore \widehat{\mathrm{n}}=\frac{\overline{\mathrm{P}}\times \overline{\mathrm{Q}}}{|\overline{\mathrm{P}}\times \overline{\mathrm{Q}}|}=\frac{5\widehat{\mathrm{i}}+5\widehat{\mathrm{j}}-5\widehat{\mathrm{k}}}{5\sqrt{3}}=\frac{\widehat{\mathrm{i}}+\widehat{\mathrm{j}}-\widehat{\mathrm{k}}}{\sqrt{3}}\end{array}$