A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill holes by entering the equation of the line of the hole.

A $20\text{ cm} \times 30\text{ cm} \times 30\text{ cm}$ cuboid is to be cut and drilled. The cuboid is positioned relative to $x$-, $y$- and $z$-axes as shown in Fig. 8.1.

\includegraphics{figure_2}

First, a plane cut is made to remove the corner at E. The cut goes through the points P, Q and R, which are the midpoints of the sides ED, EA and EF respectively.

\begin{enumerate}[label=(\roman*)]
\item Write down the coordinates of P, Q and R.

Hence show that $\overrightarrow{PQ} = \begin{pmatrix} 0 \\ 0 \\ -15 \end{pmatrix}$ and $\overrightarrow{PR} = \begin{pmatrix} -15 \\ 0 \\ 1 \end{pmatrix}$. [4]

\item Show that $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ is perpendicular to the plane through P, Q and R.

Hence find the cartesian equation of this plane. [5]
\end{enumerate}

A hole is then drilled perpendicular to triangle PQR, as shown in Fig. 8.2. The hole passes through the triangle at the point T which divides the line PS in the ratio $2:1$, where S is the midpoint of QR.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Write down the coordinates of S, and show that the point T has coordinates $(-5, 16, 25)$. [4]

\item Write down a vector equation of the line of the drill hole.

Hence determine whether or not this line passes through C. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4  Q4 [18]}}