Problem

The skew lines $l_1$ and $l_2$ have equations

$$l_1 : \mathbf{r} = (\mathbf{i} + 2\mathbf{j} - 5\mathbf{k}) + \lambda(5\mathbf{i} + \mathbf{j})$$

and

$$l_2 : \mathbf{r} = (2\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}) + \mu(8\mathbf{i} - 2\mathbf{j} + 3\mathbf{k})$$

where $\lambda$ and $\mu$ are scalar parameters.

(a) Determine a vector that is perpendicular to both $l_1$ and $l_2$.

(b) Determine an equation of the plane parallel to $l_1$ that contains $l_2$.

(c) Determine the shortest distance between $l_1$ and $l_2$.

Give your answer in simplest form.