N2024 P1 Q12

Air traffic controllers need to know the exact locations of aircraft. They use coordinates \(\left( x,y,z \right)\), with units in kilometres, to locate individual aircraft relative to the base of the control tower which is at position \(\left( 0,0,0 \right)\). The ground is horizontal and is modelled as the \(x-y\) plane.

An aircraft flies along a path with equation \(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 4 \\ 3 \\ 1 \\ \end{matrix} \\ \end{matrix} \right)+\lambda \left( \begin{matrix} \begin{matrix} 1 \\ -2 \\ 2 \\ \end{matrix} \\ \end{matrix} \right)\). The air traffic controllers are situated in the control tower \(0.1\) km above the base of the control tower.

1. Due to the atmospheric conditions at the time, the air traffic controllers can only see aircraft that come within \(4\) km of their position.
   1. Find the coordinates of this aircraft when it is at its closest point to the air traffic controllers.[4]
   2. Hence determine whether the air traffic controllers will see this aircraft.[2]

A drone flies along a path with equation

\(\frac{2x-3}{3}=y+1=\frac{z-k}{-1}\),

where \(k\) is a positive constant.

2. Determine the possible values of \({k}\) if the paths of the aircraft and the drone do not intersect.[3]
3. Find the acute angle between the paths of the aircraft and the drone.[2]
4. It is now given that \(k=2\).
   The aircraft and the drone are observed on radar to be closest to each other when they are at the points \(\left( 4,3,1 \right)\) and \(\left( 4.2,0.8,0.2 \right)\) on their respective paths.
   1. Find the distance between the aircraft and the drone at this instant.[2]
   2. By considering the vector between the aircraft and the drone at this instant, determine whether the distance found in part (d)(i) is the shortest distance between the two paths.[2]