The objective of this exercise is to study the trajectories of two submarines in the diving phase.\\
We consider that these submarines move in a straight line, each at constant speed.\\
At each instant $t$, expressed in minutes, the first submarine is located by the point $S_{1}(t)$ and the second submarine is located by the point $S_{2}(t)$ in an orthonormal reference frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ with unit metre.\\
The plane defined by $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ represents the sea surface. The $z$ coordinate is zero at sea level, negative underwater.

\begin{enumerate}
  \item We admit that, for every real $t \geqslant 0$, the point $S_{1}(t)$ has coordinates:
$$\left\{ \begin{array}{l} x(t) = 140 - 60t \\ y(t) = 105 - 90t \\ z(t) = -170 - 30t \end{array} \right.$$
a. Give the coordinates of the submarine at the beginning of the observation.\\
b. What is the speed of the submarine?\\
c. We place ourselves in the vertical plane containing the trajectory of the first submarine.

Determine the angle $\alpha$ that the submarine's trajectory makes with the horizontal plane.\\
Give the value of $\alpha$ rounded to 0.1 degree.
  \item At the beginning of the observation, the second submarine is located at the point $S_{2}(0)$ with coordinates $(68; 135; -68)$ and reaches after three minutes the point $S_{2}(3)$ with coordinates $(-202; -405; -248)$ at constant speed.\\
At what instant $t$, expressed in minutes, are the two submarines at the same depth?
\end{enumerate}