6
3
\end{array} \right) + \mu \left( \begin{array} { l }
1
0
2
\end{array} \right)$$
Find the acute angle between the lines.
4 A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill boles 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 cut 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]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-3_414_740_460_302}
\captionsetup{labelformat=empty}
\caption{Fig. 8.1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-3_449_737_425_1062}
\captionsetup{labelformat=empty}
\caption{Fig. 8.2}
\end{figure}
First, a plane cut 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 \(\boldsymbol { 0 } \mathbf { F } \mathrm { Q }\) and \(\mathrm { R } \left( \begin{array} { l } F \\ 1 \end{array} \right]\)
Hence show that \(\mathrm { PQ } = { } _ { - }\): and \(\mathrm { PR } =\)
(U) Show th,i tho \(, 0010,11\) is pc,pondio,la, to the pl'ute through \(P , Q\) rudd \(R\)
Hence find the cartesian equation of this plane.
A hole is then drilled perpendicular to triangle PQR , as shown in Fig. 82. The hole passes through the triangle at the point T which divides the line PS in the ratio 2 : I , where S is the midpoint of QR . - Write down the coordinates of S , and show that the point T has coordinates \(( - 5.16 \mathrm { i } , 25 )\).
- Write down a vector equation of the line of the drill hole.
Hence determine whether or not this line passes through C .
5 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The Oxy plane is horizontal.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{253ddd65-d92b-46ce-bf17-b4f6e3d32ec0-4_555_1004_486_565}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{figure} - Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
- Show that the vector \(\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\) is normal to the plane through \(\mathrm { A } , \mathrm { D }\) and E .
Hence find the equation of this plane. Given that B lies in this plane, find \(a\).
- Verify that the equation of the plane BCD is \(x + z = 8\).
Hence find the acute angle between the planes ABDE and BCD .