Two aircraft are approaching an airport. We equip space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ whose origin O is the base of the control tower, and the ground is the plane $P_0$ with equation $z = 0$. The unit of the axes corresponds to 1 km. We model the aircraft as points.

Aircraft Alpha transmits to the tower its position at $\mathrm{A}(-7; 1; 7)$ and its trajectory is directed by the vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}$.

Aircraft Beta transmits a trajectory defined by the line $d_{\mathrm{B}}$ passing through point B with a parametric representation:
$$\left\{\begin{aligned} x &= -11 + 5t \\ y &= -5 + t \\ z &= 11 - 4t \end{aligned}\right. \text{ where } t \text{ describes } \mathbb{R}.$$

\begin{enumerate}
  \item If it does not deviate from its trajectory, determine the coordinates of point S where aircraft Beta will touch the ground.
  \item a. Determine a parametric representation of the line $d_{\mathrm{A}}$ characterizing the trajectory of aircraft Alpha.\\
b. Can the two aircraft collide?
  \item a. Prove that aircraft Alpha passes through position $\mathrm{E}(-3; -1; 1)$.\\
b. Justify that a Cartesian equation of the plane $P_{\mathrm{E}}$ passing through E and perpendicular to the line $d_{\mathrm{A}}$ is:
$$2x - y - 3z + 8 = 0.$$
c. Verify that the point $\mathrm{F}(-1; -3; 3)$ is the intersection point of the plane $P_{\mathrm{E}}$ and the line $d_{\mathrm{B}}$.\\
d. Calculate the exact value of the distance EF, then verify that this corresponds to a distance of 3464 m, to the nearest 1 m.
  \item Air traffic regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at E and F at the same instant, is their safety distance respected?
\end{enumerate}