What is the shortest distance between $\vec{r} = (\hat{i} + \hat{j}) + \lambda(\hat{i} + \hat{k})$ and $\vec{r} = (3\hat{j} + 5\hat{k}) + \mu(\hat{j} + \hat{k})$?

Explanation

1. 01

   $\vec{b_1} \times \vec{b_2} = -\hat{i} - \hat{j} + \hat{k}$, of length $\sqrt{3}$, and $\vec{a_2} - \vec{a_1} = -\hat{i} + 2\hat{j} + 5\hat{k}$.

2. 02

   $d = \frac{\lvert 1 - 2 + 5 \rvert}{\sqrt{3}} = \frac{4}{\sqrt{3}}$.