Write the vector equation of each of the following lines and hence determine the distance between them:

Write the vector equation of each of the following lines and hence determine the distance between them:

$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $\frac{x-3}{4}=\frac{y-3}{6}=\frac{z+5}{12}$

HINT: The given lines are

$\mathrm{L}_{1}: \overrightarrow{\mathrm{r}}=(-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}})+\lambda(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$

$\mathrm{L}_{2}: \overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})+2 \mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$

Now, find the distance between the parallel lines $L_{1}$ and $L_{2}$.