What is the shortest distance between the $x$-axis and the line through $(0, 0, 3)$ parallel to the $y$-axis?

Explanation

1. 01

   $\vec{b}_1 \times \vec{b}_2 = \hat{i} \times \hat{j} = \hat{k}$, of length 1, and $\vec{a}_2 - \vec{a}_1 = 3\hat{k}$.

2. 02

   $d = \lvert 3\hat{k} \cdot \hat{k} \rvert = 3$: the common perpendicular runs along the $z$-axis from $(0, 0, 0)$ to $(0, 0, 3)$.