What is the shortest distance between the line through $A(1, -1, 0)$ and $B(2, 1, 2)$ and the line through $C(3, 0, 1)$ and $D(5, 1, -1)$?

Explanation

1. 01

   $\vec{b}_1 = \overrightarrow{AB} = \hat{i} + 2\hat{j} + 2\hat{k}$ and $\vec{b}_2 = \overrightarrow{CD} = 2\hat{i} + \hat{j} - 2\hat{k}$; $\vec{b}_1 \times \vec{b}_2 = -6\hat{i} + 6\hat{j} - 3\hat{k}$, of length 9.

2. 02

   $\overrightarrow{AC} = 2\hat{i} + \hat{j} + \hat{k}$, so $d = \frac{\lvert -12 + 6 - 3 \rvert}{9} = 1$.