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

Explanation

1. 01

   $-4\hat{i} + 2\hat{j} - 4\hat{k} = -2(2\hat{i} - \hat{j} + 2\hat{k})$: the lines are parallel, so use one direction $\vec{b} = 2\hat{i} - \hat{j} + 2\hat{k}$, with $\lvert \vec{b} \rvert = 3$.

2. 02

   $\vec{b} \times (\vec{a}_2 - \vec{a}_1) = (2\hat{i} - \hat{j} + 2\hat{k}) \times (2\hat{i} - 3\hat{j} + 2\hat{k}) = 4\hat{i} - 4\hat{k}$, of length $4\sqrt{2}$.

3. 03

   $d = \frac{4\sqrt{2}}{3}$.