Problem

The plane $\Pi_1$ has equation

$$\mathbf{r} = \begin{pmatrix} 2 \\ 5 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 5 \\ -6 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix}$$

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

(a) Determine

$$(7\mathbf{i}+5\mathbf{j}-6\mathbf{k}) \times (-3\mathbf{i}+\mathbf{j}+2\mathbf{k})$$

(b) Hence show that the equation of $\Pi_1$ can be written in the form

$$\mathbf{r}\cdot(8\mathbf{i}+2\mathbf{j}+11\mathbf{k})=p$$

where $p$ is a constant to be determined.

Given that

• the plane $\Pi_2$ has equation $x-y+z=7$
• the planes $\Pi_1$ and $\Pi_2$ intersect in the line $l_1$

(c) determine an equation for $l_1$ giving your answer in the form $(\mathbf{r}-\mathbf{a})\times \mathbf{b}=0$ where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors.

Given also that

• the point A has coordinates $(2,1,3)$
• the point B has coordinates $(3,0,2)$
• the line $l_2$ passes through A and B

(d) determine the shortest distance between $l_1$ and $l_2$