Problem

The lines $l_1$ and $l_2$ have equations

$$l_1 : \mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{k})$$

and

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

where $s$ is a constant and $\lambda$ and $\mu$ are scalar parameters.

Given that $l_1$ and $l_2$ both lie in a common plane $\Pi_1$,

(a) show that an equation for $\Pi_1$ is $3x + y - z = 3$.

(b) find the value of $s$.

The plane $\Pi_2$ has equation

$$\mathbf{r}\cdot(\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 3.$$

(c) Find an equation for the line of intersection of $\Pi_1$ and $\Pi_2$.

(d) Find the acute angle between $\Pi_1$ and $\Pi_2$, giving your answer in degrees to 3 significant figures.