Which is the vector equation of the line through $(-1, 2, 1)$ perpendicular to both $\frac{x - 3}{1} = \frac{y + 1}{2} = \frac{z - 2}{-1}$ and $\frac{x}{2} = \frac{y - 4}{1} = \frac{z + 1}{1}$?

Explanation

1. 01

   A direction perpendicular to both is $\vec{b}_1 \times \vec{b}_2 = (\hat{i} + 2\hat{j} - \hat{k}) \times (2\hat{i} + \hat{j} + \hat{k}) = 3\hat{i} - 3\hat{j} - 3\hat{k}$, or $\hat{i} - \hat{j} - \hat{k}$.

2. 02

   Through $(-1, 2, 1)$: $\vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} - \hat{k})$.