The distance between the parallel lines $\overline{r}=2\overline{i}+3\overline{j}-\overline{k}+\lambda (\overline{i}-\overline{j}+2\overline{k})$ and $\overline{r}=-3\overline{i}+4\overline{j}+\overline{k}+\mu (\overline{i}-\overline{j}+2\overline{k})$ is

${\overline{a}}_1=2\overline{i}+3\overline{j}-\overline{k}$, ${\overline{a}}_2=-3\overline{i}+4\overline{j}+\overline{k}$, $\overline{b}=\overline{i}-\overline{j}+2\overline{k}$, ${\overline{a}}_1-{\overline{a}}_2=5\overline{i}-\overline{j}-2\overline{k}$

$\overline{b}\times ({\overline{a}}_1-{\overline{a}}_2)=\left|\begin{array}{ccc}\overline{i}& \overline{j}& \overline{k}\\ 1& -1& 2\\ 5& -1& -2\end{array}\right|=4\overline{i}+12\overline{j}+4\overline{k}$

$\left|\overline{b}\times ({\overline{a}}_1-{\overline{a}}_2)\right|=\sqrt{176}$, $\left|\overline{b}\right|=\sqrt{6}$

Distance = $\frac{\left|\overline{b}\times ({\overline{a}}_1-{\overline{a}}_2)\right|}{\left|\overline{b}\right|}=\frac{\sqrt{176}}{\sqrt{6}}=2\sqrt{\frac{22}{3}}$