The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres.

\includegraphics{figure_1}

Relative to axes $Ox$ (due east), $Oy$ (due north) and $Oz$ (vertically upwards), the coordinates of the points are as follows.

A: $(0, 0, -15)$ \quad B: $(100, 0, -30)$ \quad C: $(0, 100, -25)$
D: $(0, 0, -40)$ \quad E: $(100, 0, -50)$ \quad F: $(0, 100, -35)$

\begin{enumerate}[label=(\roman*)]
\item Verify that the cartesian equation of the plane ABC is $3x + 2y + 20z + 300 = 0$. [3]

\item Find the vectors $\overrightarrow{DE}$ and $\overrightarrow{DF}$. Show that the vector $2\mathbf{i} - \mathbf{j} + 20\mathbf{k}$ is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF. [6]

\item By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. [4]
\end{enumerate}

It is decided to drill down to the seam from a point R $(15, 34, 0)$ in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Write down a vector equation of the line RS.

Find the coordinates of S. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4  Q1 [18]}}