The shortest distance between lines \(\bar{r}=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})\) and \(\bar{r}=(2 \hat{i}-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})\) is

\(\begin{aligned} & \overline{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}}) \text { and } \\ & \overline{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}) \\ & \text { Let } \mathrm{a}_1=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \mathrm{a}_2=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}} \\ & \mathrm{b}_1=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}}, \mathrm{b}_2=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}} \\ & \overline{\mathrm{a}_2}-\overline{\mathrm{a}_1}=\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \\ & \overline{\mathrm{b}}_1 \times \overline{\mathrm{b}}_2=-2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} \\ & \left|\overline{\mathrm{b}}_1 \times \overline{\mathrm{b}}_2\right|=\sqrt{38}\end{aligned}\)
\(\begin{aligned} \text { Shortest distance } & =\left|\frac{\left(\overline{\mathrm{b}}_1 \times \overline{\mathrm{b}}_2\right) \cdot\left(\overline{\mathrm{a}}_2-\overline{\mathrm{a}}_1\right)}{\left|\overline{\mathrm{b}}_1 \times \overline{\mathrm{b}}_2\right|}\right| \\ & =\left|\frac{4}{\sqrt{38}}\right|=\left|\frac{2 \sqrt{2}}{\sqrt{19}}\right| \text { units }\end{aligned}\)