For which $k$ do the lines $\frac{x - 2}{1} = \frac{y + 1}{2} = \frac{z - 3}{-1}$ and $\frac{x - 4}{2} = \frac{y - k}{1} = \frac{z - 1}{1}$ intersect?

Explanation

1. 01

   The lines are not parallel, so they meet exactly when $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$.

2. 02

   $\vec{a}_2 - \vec{a}_1 = 2\hat{i} + (k + 1)\hat{j} - 2\hat{k}$ and $\vec{b}_1 \times \vec{b}_2 = 3\hat{i} - 3\hat{j} - 3\hat{k}$.

3. 03

   $6 - 3(k + 1) + 6 = 9 - 3k = 0$, so $k = 3$.