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

Explanation

1. 01

   $-4\hat{i} + 6\hat{j} - 2\hat{k} = -2(2\hat{i} - 3\hat{j} + \hat{k})$: the directions are proportional.

2. 02

   $\cos \theta = \frac{\lvert -8 - 18 - 2 \rvert}{\sqrt{14}\sqrt{56}} = \frac{28}{28} = 1$, so $\theta = 0$: the lines are parallel.