Let L 1 (respectively L 2 ) be the line passing through 2 i ^ - k ^ (respectively 2 i ^ + j ^ - 3 k ^ ) and…

Given that line $L_1$ passes through $2\hat{i}-\hat{k}$ and parallel to $3\hat{i}-\hat{j}+2\hat{k}$

and line $L_2$ passes through $2\hat{i}+\hat{j}-3\hat{k}$ and parallel to $\hat{i}-2\hat{j}+\hat{k}$.

Then, $a_1=2\hat{i}-\hat{k}$, $b_1=3\hat{i}-\hat{j}+2\hat{k}$ and $a_2=2\hat{i}+\hat{j}-3\hat{k}$, $b_2=\hat{i}-2\hat{j}+\hat{k}$

Shortest distance between lines is given by, $d=\frac{\left({\overrightarrow{b}}_1\times {\overrightarrow{b}}_2\right)·\left({\overrightarrow{a}}_2-{\overrightarrow{a}}_1\right)}{\left|{\overrightarrow{b}}_1\times {\overrightarrow{b}}_2\right|}$.

Now, $\left({\overrightarrow{b}}_1\times {\overrightarrow{b}}_2\right)=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& -1& 2\\ 1& -2& 1\end{array}\right|=3\hat{i}-\hat{j}-5\hat{k}$

$\Rightarrow \left|{\overrightarrow{b}}_1\times {\overrightarrow{b}}_2\right|=\sqrt{3^2+{\left(-1\right)}^2+{\left(-5\right)}^2}=\sqrt{35}$

And, $\left({\overrightarrow{a}}_2-{\overrightarrow{a}}_1\right)=\hat{j}-2\hat{k}$

Putting these values, we get $d=\frac{\left(3\hat{i}-\hat{j}-5\hat{k}\right)·\left(\hat{j}-2\hat{k}\right)}{\sqrt{35}}=\frac{-1+10}{\sqrt{35}}=\frac{9}{\sqrt{35}}$