4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{55803551-f13d-419f-8b51-31642bd20b6a-12_474_1063_264_502}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A small aircraft is landing in a field.
In a model for the landing the aircraft travels in different straight lines before and after it lands, as shown in Figure 2.
The vector \(\mathbf { v } _ { \mathbf { A } }\) is in the direction of travel of the aircraft as it approaches the field.
The vector \(\mathbf { V } _ { \mathbf { L } }\) is in the direction of travel of the aircraft after it lands.
With respect to a fixed origin, the field is modelled as the plane with equation
$$x - 2 y + 25 z = 0$$
and
$$\mathbf { v } _ { \mathbf { A } } = \left( \begin{array} { r }
3 \\
- 2 \\
- 1
\end{array} \right)$$
- Write down a vector \(\mathbf { n }\) that is a normal vector to the field.
- Show that \(\mathbf { n } \times \mathbf { v } _ { \mathbf { A } } = \lambda \left( \begin{array} { r } 13 \\ 19 \\ 1 \end{array} \right)\), where \(\lambda\) is a constant to be determined.
When the aircraft lands it remains in contact with the field and travels in the direction \(\mathbf { v } _ { \mathbf { L } }\) The vector \(\mathbf { v } _ { \mathbf { L } }\) is in the same plane as both \(\mathbf { v } _ { \mathbf { A } }\) and \(\mathbf { n }\) as shown in Figure 2.
- Determine a vector which has the same direction as \(\mathbf { V } _ { \mathbf { L } }\)
- State a limitation of the model.