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

Explanation

1. 01

   $\vec{b_1} \times \vec{b_2} = -3\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{a_2} - \vec{a_1} = -2\hat{j} + \hat{k}$; their dot product is $0$.

2. 02

   So the distance is 0: the lines meet, at $(2, 1, 5)$ where $\lambda = \mu = 1$.