Measured from the origin $O$, point $A$ has position vector $\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}$. Line $l$ is defined by $\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})$.
(i)[5]
Find the position vector of the foot of the perpendicular from $A$ to $l$. Hence find the position vector of the reflection of $A$ in $l$.
(ii)[3]
Find the equation of the plane through the origin that contains $l$. Give your answer in the form $ax + by + cz = d$.
(iii)[3]
Find the exact value of the perpendicular distance of $A$ from this plane.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Find $\overrightarrow{AP}$ for any point chosen on $l$” …