2 A piece of cloth ABDC is attached to the tops of vertical poles \(\mathrm { AE } , \mathrm { BF } , \mathrm { DG }\) and CH , where \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b46db958-aa88-47fb-8db3-786472791577-2_916_1255_464_397}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{figure}
- Write down the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\). Hence calculate the angle BAC .
- Verify that the equation of the plane ABC is \(x + y - 2 z + d = 0\), where \(d\) is a constant to be determined.
Calculate the acute angle the plane makes with the horizontal plane.
- Given that \(\mathrm { A } , \mathrm { B } , \mathrm { D }\) and C are coplanar, show that \(k = 3\).
Hence show that ABDC is a trapezium, and find the ratio of CD to AB .