\textbf{Exercise 2 -- Common to all candidates}

Alex and Élisa, two drone pilots, are training on a terrain consisting of a flat part bordered by an obstacle. We consider an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), with one unit corresponding to ten metres. Six points are defined by their coordinates:
$$\mathrm { O } ( 0 ; 0 ; 0 ) , \mathrm { P } ( 0 ; 10 ; 0 ) , \mathrm { Q } ( 0 ; 11 ; 1 ) , \mathrm { T } ( 10 ; 11 ; 1 ) , \mathrm { U } ( 10 ; 10 ; 0 ) \text { and } \mathrm { V } ( 10 ; 0 ; 0 )$$
The flat part is delimited by the rectangle OPUV and the obstacle by the rectangle PQTU.

The two drones are assimilable to two points and follow rectilinear trajectories:
\begin{itemize}
  \item Alex's drone follows the trajectory carried by the line $( \mathrm { AB } )$ with $\mathrm { A } ( 2 ; 4 ; 0.25 )$ and $\mathrm { B } ( 2 ; 6 ; 0.75 )$;
  \item Élisa's drone follows the trajectory carried by the line (CD) with C(4; 6; 0.25) and D(2; 6; 0.25).
\end{itemize}

\textbf{Part A: Study of Alex's drone trajectory}
\begin{enumerate}
  \item Determine a parametric representation of the line ( AB ).
  \item \begin{enumerate}
    \item[a.] Justify that the vector $\vec { n } ( 0 ; 1 ; - 1 )$ is a normal vector to the plane (PQU).
    \item[b.] Deduce a Cartesian equation of the plane (PQU).
  \end{enumerate}
  \item Prove that the line (AB) and the plane (PQU) intersect at the point I with coordinates $\left( 2 ; \frac { 37 } { 3 } ; \frac { 7 } { 3 } \right)$.
  \item Explain why, following this trajectory, Alex's drone does not encounter the obstacle.
\end{enumerate}

\textbf{Part B: Minimum distance between the two trajectories}

To avoid a collision between their two devices, Alex and Élisa impose a minimum distance of 4 metres between the trajectories of their drones. For this, we consider a point $M$ on the line (AB) and a point $N$ on the line (CD). There then exist two real numbers $a$ and $b$ such that $\overrightarrow { \mathrm { A } M } = a \overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { C } N } = b \overrightarrow { \mathrm { CD } }$.

\begin{enumerate}
  \item Prove that the coordinates of the vector $\overrightarrow { M N }$ are $( 2 - 2 b ; 2 - 2 a ; - 0.5 a )$.
  \item It is admitted that the lines (AB) and (CD) are not coplanar. It is also admitted that the distance $MN$ is minimal when the line ( $MN$ ) is perpendicular to both the line ( AB ) and the line (CD). Prove then that the distance $MN$ is minimal when $a = \frac { 16 } { 17 }$ and $b = 1$.
  \item Deduce the minimum value of the distance $MN$ and conclude.
\end{enumerate}