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

Explanation

1. 01

   The $z$-axis is $\vec{r} = \mu\hat{k}$, so $\vec{b}_1 \times \vec{b}_2 = (2\hat{i} + \hat{j} - \hat{k}) \times \hat{k} = \hat{i} - 2\hat{j}$, of length $\sqrt{5}$.

2. 02

   $\vec{a}_2 - \vec{a}_1 = -3\hat{i} + \hat{j} - 2\hat{k}$, so $d = \frac{\lvert -3 - 2 \rvert}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}$.