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

Explanation

1. 01

   $\vec{b}_1 \times \vec{b}_2 = -3\hat{i} + 3\hat{k}$, with magnitude $3 \sqrt{2}$; $\vec{a}_2 - \vec{a}_1 = \hat{i} - 3\hat{j} - 4\hat{k}$.

2. 02

   $d = \frac{\lvert (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) \rvert}{\lvert \vec{b}_1 \times \vec{b}_2 \rvert} = \frac{15}{3 \sqrt{2}} = \frac{5 \sqrt{2}}{2}$.