For $L_1: \vec{r} = (\hat{i} - \hat{k}) + \lambda(2\hat{i} + 2\hat{j} + \hat{k})$ and $L_2: \vec{r} = (2\hat{j} + 3\hat{k}) + \mu(\hat{i} - 2\hat{j} + 2\hat{k})$, which of the following statements are true?

I. $L_1$ and $L_2$ are perpendicular.

II. $L_1$ and $L_2$ intersect.

III. The shortest distance between $L_1$ and $L_2$ is 4.

IV. $\vec{b}_1 \times \vec{b}_2 = 6\hat{i} + 3\hat{j} - 6\hat{k}$.

Explanation

1. 01

   $\vec{b}_1 \cdot \vec{b}_2 = 2 - 4 + 2 = 0$, so I is true; $\vec{b}_1 \times \vec{b}_2 = 6\hat{i} - 3\hat{j} - 6\hat{k}$, so IV is false.

2. 02

   $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = (-\hat{i} + 2\hat{j} + 4\hat{k}) \cdot (6\hat{i} - 3\hat{j} - 6\hat{k}) = -36 \ne 0$: the lines do not meet, so II is false, and $d = \frac{36}{9} = 4$, so III is true.