The equations for the two lines $l$ and $m$ are $\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$, in that order.
(i)[3]
Show that the lines do not intersect.
(ii)[3]
Calculate the acute angle made by the directions of the lines.
(iii)[5]
Find the equation of the plane which passes through the point $(3, -2, -1)$ and is parallel to both $l$ and $m$. Give your answer in the form $ax + by + cz = d$.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Set equal at least two corresponding component pairs for general points on $l$ and $m$, then solve for $\lambda$ or $\mu$” …