The position vectors of points $A$, $B$ and $C$, measured from the origin $O$, are given by $\vec{OA} = \begin{pmatrix}1\\2\\0\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}3\\0\\1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}1\\1\\4\end{pmatrix}$. Plane $m$ is perpendicular to $AB$ and passes through point $C$.
(i)[2]
Find a vector equation for the line that passes through $A$ and $B$.
(ii)[2]
Obtain the equation of plane $m$, giving your answer in the form $ax + by + cz = d$.
(iii)[5]
The line through $A$ and $B$ meets plane $m$ at 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 write a vector equation for $AB$” …