A passage of an aerial acrobatics show in a duo is modelled as follows:
\begin{itemize}
  \item we place ourselves in an orthonormal coordinate system $(O ; \vec { \imath } , \vec { \jmath } , \vec { k })$, where one unit represents one metre;
  \item plane no. 1 must travel from point O to point $A(0 ; 200 ; 0)$ along a straight trajectory, at the constant speed of $200 \mathrm {~m/s}$;
  \item plane no. 2 must travel from point $B(-33 ; 75 ; 44)$ to point $C(87 ; 75 ; -116)$ also along a straight trajectory, and at the constant speed of $200 \mathrm {~m/s}$;
  \item at the same instant, plane no. 1 is at point O and plane no. 2 is at point B.
\end{itemize}

\begin{enumerate}
  \item Justify that plane no. 2 will take the same time to travel segment $[BC]$ as plane no. 1 to travel segment $[OA]$.
  \item Show that the trajectories of the two planes intersect.
  \item Is there a risk of collision between the two planes during this passage?
\end{enumerate}