The shortest distance between the lines through the points (2, 3, 1), (4, 5, 2) and parallel to the vectors (3, 4, 2), (4, 5, 3) respectively is

$$\begin{gathered} {\overrightarrow{L}}_1=(2\widehat{i}+3\widehat{j}+\widehat{k})+\lambda (3\widehat{i}+4\widehat{j}+2\widehat{k}) \\ \overrightarrow{L_2}=(4\widehat{i}+5\widehat{j}+2\widehat{k})+\mu (4\widehat{i}+5\widehat{j}+3\widehat{k}) \end{gathered}$$

$$\begin{gathered} \therefore \text{ Shortest distance }=\left|\frac{\left(\overrightarrow{a_1}-\overrightarrow{a_2}\right)\cdot \left(\overrightarrow{b_1}\times \overrightarrow{b_2}\right)}{\left|\overrightarrow{b_1}\times \overrightarrow{b_2}\right|}\right| ···\left(i\right) \\ {\overrightarrow{a}}_2-{\overrightarrow{a}}_1=\left[\right(4\widehat{i}+5\widehat{j}+2\widehat{k})-(2\widehat{i}+3\widehat{j}+\widehat{k}\left)\right] \\ \Rightarrow \overrightarrow{a_2}-{\overrightarrow{a}}_2=[2\widehat{i}+2\widehat{j}+\widehat{k}] ···\left(ii\right) \\ {\overrightarrow{b}}_1\times {\overrightarrow{b}}_2=(3\widehat{i}+4\widehat{k}+2\widehat{j})\times (4\widehat{i}+5\widehat{k}+3\widehat{k}) \\ =\left|\begin{array}{l}\widehat{i}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\widehat{j}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\widehat{k}\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\\ 4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\end{array}\right| \\ =\widehat{i}(12-10)-\widehat{j}(9-6)+\widehat{k}(15-16) \\ \overrightarrow{b_1}\times \overrightarrow{b_2}=2\widehat{i}-\widehat{j}-\widehat{k} ···\left(iii\right) \\ \left|\overrightarrow{b_1}\times \overrightarrow{b_2}\right|=\sqrt{(2{)}^2+(-1{)}^2+(-1{)}^2} \\ =\sqrt{4+1+1} \\ \left|{\overrightarrow{b}}_1\times {\overrightarrow{b}}_2\right|=\sqrt{6} ···\left(iv\right) \\ \text{ Shortest distance }=\left|\frac{(2\widehat{i}+2\widehat{j}+\widehat{k})\cdot (2\widehat{i}-\widehat{j}-\widehat{k})}{\sqrt{6}}\right| \\ \text{ Shortest distance }=\left|\frac{4-2-1}{\sqrt{6}}\right| \\ \text{ Shortest distance }=\frac{\left|1\right|}{\sqrt{6}} \end{gathered}$$