The distance between parallel lines \(\overline{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})\) and \(\overline{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})\) is

The distance between given parallel lines
\(=\left|\frac{[(\hat{i}-2 \hat{i})+(-\hat{j}+\hat{j})+(2 \hat{k}-\hat{k})] \times(2 \hat{i}+\hat{j}-2 \hat{k})}{|2 \hat{i}+\hat{j}-2 \hat{k}|}\right|=\) \(\left|\frac{(-\hat{i}+\hat{k}) \times(2 \hat{i}+\hat{j}-2 \hat{k})}{\sqrt{(2)^2+(1)^2+(-2)^2}}\right|\)
\((-\hat{ i }+\hat{ k }) \times(2 \hat{ i }+\hat{ j }-2 \hat{ k })=\left|\begin{array}{ccc} i & \hat{ j } & \hat{ k } \\ -1 & 0 & 1 \\ 2 & 1 & -2\end{array}\right|=\) \(\hat{ i }(0-1)-\hat{ j }(2-2)+\hat{ k }(-1-0)=-\hat{ i }-\hat{ k }\)
\(\therefore d =\left|\frac{\mid-\hat{ i }-\hat{ k }}{\sqrt{9}}\right|=\frac{\sqrt{2}}{3}\) units