The distance between the parallel lines \(\frac{x-2}{2}=\frac{y-4}{5}=\frac{z-1}{2}\) and \(\frac{x-1}{3}=\frac{y+1}{5}=\frac{z+3}{2}\) is

Let \(\overline{\mathrm{a}}_1=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overline{\mathrm{a}}_2=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}\)
\(\therefore \overline{\mathrm{a}}_2-\overline{\mathrm{a}}_1=-\hat{\mathrm{i}}-6 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}\) and let \(\overline{\mathrm{b}}=3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\)
\(\overline{\text b }\left|\overline{\text a }_2-\overline{\text a }_1\right|=\left|\begin{array}{ccc}\hat{\text i } & \hat{\text j } & \hat{\text k } \\ 3 & 5 & 2 \\ -1 & -6 & -4\end{array}\right|=\hat{\text i }(-20+12)-\) \(\hat{\text j }(-12+2)+\hat{\text k }(-18+5)\)
\(=-8 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}-13 \hat{\mathrm{k}}\)
\(\therefore\left|\overline{\mathrm{b}} \times\left(\overline{\mathrm{a}}_2-\overline{\mathrm{a}}_1\right)\right|=\sqrt{64+100+169}=\sqrt{133}
\)
Also \(|\overline{\mathrm{b}}|=\sqrt{9+25+4}=\sqrt{38}\)
\(\therefore\) Distance between given lines \(=\sqrt{\frac{333}{38}}\) units