What is the distance between the parallel lines $\vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$ and $\vec{r} = (3\hat{i} + 3\hat{j} - 5\hat{k}) + \mu(2\hat{i} + 3\hat{j} + 6\hat{k})$?

Explanation

1. 01

   $\vec{a}_2 - \vec{a}_1 = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} \times (\vec{a}_2 - \vec{a}_1) = -9\hat{i} + 14\hat{j} - 4\hat{k}$.

2. 02

   $d = \frac{\lvert \vec{b} \times (\vec{a}_2 - \vec{a}_1) \rvert}{\lvert \vec{b} \rvert} = \frac{\sqrt{293}}{7} = \frac{\sqrt{293}}{7}$.