4 A computer-controlled machine can be programmed to make ats 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 \(20 \mathrm {~cm} \times 30 \mathrm {~cm} \times 30 \mathrm {~cm}\) cuboid is to be at and drilled. The cuboid is positioned relative to \(x\)-, \(y ^ { 2 }\) and z-axes as shown in Fig. 8.1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7a52b6ce-a0cc-421d-8eae-3b6cf098e381-4_416_702_463_322}
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\caption{Fig.8.1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7a52b6ce-a0cc-421d-8eae-3b6cf098e381-4_420_683_459_1044}
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\caption{Fig. 8.2}
\end{figure}
Fkst, a plane out is made to remove the comer at \(E\). The cut goes through the points \(P . Q\) and \(R\), which are the midpoints of the sides \(\mathrm { ED } , \mathrm { EA }\) and EF respectively.
- Write down the coordinates of \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\).
$$\text { Henceshowlhat } \mathbb { N } ^ { 1 } \left\{ \begin{array} { l }
1 \\
0 \\
0
\end{array} \right\} \text { and } \left\{ \begin{array} { l }
1 \\
0
\end{array} \right\}$$
(;) Show that the veeto, \(\binom { \text { I } } { \text { ) } }\) is pc,pcndicula, to the plone through \(\mathrm { P } , \mathrm { Q }\), nd R
Hence find the Cartesian equation of this plane.
A hole is then drilled perpendicular to lriangle \(P Q R\), as shown in Fig. 82. The hole passes through the triangle at the point \(T\) which divides the line \(P S\) in the ratio 2 : \(I\), where \(S\) is the midpoint of \(Q R\). - Write down the coordinates of S , and show that the point T has coordinates \(( - 5,16,25 )\).
- Write down a vector equation of the line of the drill hole.
Hence determine whether or not this line passes through C .