$$\begin{gathered} \mathrm{lf} \overline{\mathrm{a}} = 2\overline{\mathrm{i}}+\overline{\mathrm{j}}-3\overline{\mathrm{k}} , \overline{\mathrm{b}}=\overline{\mathrm{i}} -2\overline{\mathrm{j}} +\overline{\mathrm{k}}, \mathrm{then} \mathrm{a} \mathrm{vector} \\ \mathrm{of} \mathrm{length} 5 \mathrm{and} \mathrm{perpendicular} \mathrm{to} \mathrm{both} \overline{\mathrm{a}} \mathrm{and} \overline{\mathrm{b}} \mathrm{is} \end{gathered}$$

$\begin{array}{rcl}\overline{\mathrm{a}}\times \overline{\mathrm{b}}& =& \left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& -3\\ 1& -2& 1\end{array}\right|=\mathrm{i}(-5)-\mathrm{j}\left(5\right)+\mathrm{k}(-5)\\ \overline{\mathrm{a}}\times \overline{\mathrm{b}}& =& 5(-\mathrm{i}-\mathrm{j}-\mathrm{k}) \Rightarrow |\overline{\mathrm{a}}\times \overline{\mathrm{b}}|=5\sqrt{3}\end{array}$

a vector length 5 and perpendicular to $\overline{\mathrm{a}} \& \overline{\mathrm{b}}$

$$\begin{gathered} =5\left(\frac{\overline{\mathrm{a}}\times \overline{\mathrm{b}}}{|\overline{\mathrm{a}}\times \overline{\mathrm{b}}|}\right)=5\left(\frac{5(-\mathrm{ij}-\mathrm{k})}{5\sqrt{3}}\right) \\ =\pm \frac{5}{\sqrt{3}} (\mathrm{i}+\mathrm{j}+\mathrm{k}) \end{gathered}$$