7 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(\mathrm { O } x y\) plane is horizontal.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9bceee25-35bd-448b-a4a2-1a5667be5f11-03_547_987_1580_539}
\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 .