What is the perpendicular distance of the point $(5, 1, 5)$ from the line $\vec{r} = (\hat{i} - \hat{j}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})$?

Explanation

1. 01

   $\vec{p} - \vec{a} = 4\hat{i} + 2\hat{j} + 5\hat{k}$ and $\vec{b} \times (\vec{p} - \vec{a}) = -12\hat{i} - 6\hat{j} + 12\hat{k}$, of length 18.

2. 02

   $d = \frac{18}{\lvert \vec{b} \rvert} = \frac{18}{3} = 6$.