9758/2021/P1/Q08

The lines $l_1$ and $l_2$ have equations

respectively, where $\lambda$ and $\mu$ are parameters.
(a) Find a cartesian equation of the plane containing $l_1$ and the point $(1,-3,-1)$.

[4]

(b) Show that $l_1$ is perpendicular to $l_2$.

[2]

(c) (i) Find values of $\lambda$ and $\mu$ such that $\mathbf{r}_1-\mathbf{r}_2$ is perpendicular to both $l_1$ and $l_2$. State the position vectors of the points where the common perpendicular meets $l_1$ and $l_2$.

[6]

(ii) Find the length of this common perpendicular.

[2]