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.

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.

Determine the possible values of ${k}$ if the paths of the aircraft and the drone do not intersect.
[3]
Find the acute angle between the paths of the aircraft and the drone.
[2]
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]

2024 TYS

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