Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix}1\\2\\0\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}3\\0\\1\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}1\\1\\4\end{pmatrix}$. The plane $m$ is perpendicular to $AB$ and passes through the point $C$.
(i)[2]
Find a vector equation for the line through $A$ and $B$.
(ii)[2]
Obtain the equation of the plane $m$, and present it in the form $ax + by + cz = d$.
(iii)[5]
The line through $A$ and $B$ meets the plane $m$ at the point $N$. Find the position vector of $N$ and show that $CN = \sqrt{13}$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use the correct method to obtain a vector equation for $AB$” …