In space equipped with an orthonormal coordinate system, we consider:
\begin{itemize}
  \item the points $\mathrm { A } ( 0 ; 1 ; - 1 )$ and $\mathrm { B } ( - 2 ; 2 ; - 1 )$.
  \item the line $\mathscr { D }$ with parametric representation $\left\{ \begin{array} { r l } x & = - 2 + t \\ y & = 1 + t \\ z & = - 1 - t \end{array} , t \in \mathbb { R } \right.$.
\end{itemize}

\begin{enumerate}
  \item Determine a parametric representation of the line ( AB ).
  \item a. Show that the lines $( \mathrm { AB } )$ and $\mathscr { D }$ are not parallel.\\
b. Show that the lines $( \mathrm { AB } )$ and $\mathscr { D }$ are not intersecting.
\end{enumerate}

In the following, the letter $u$ denotes a real number.\\
We consider the point $M$ of the line $\mathscr { D }$ with coordinates ( $- 2 + u ; 1 + u ; - 1 - u$ ).\\
3. Verify that the plane $\mathscr { P }$ with equation $x + y - z - 3 u = 0$ is orthogonal to the line $\mathscr { D }$ and passes through the point $M$.\\
4. Show that the plane $\mathscr { P }$ and the line (AB) intersect at a point $N$ with coordinates $( - 4 + 6 u ; 3 - 3 u ; - 1 )$.\\
5. a. Show that the line $( M N )$ is perpendicular to the line $\mathscr { D }$.\\
b. Does there exist a value of the real number $u$ for which the line ( $M N$ ) is perpendicular to the line $( \mathrm { AB } )$ ?\\
6. a. Express $M N ^ { 2 }$ as a function of $u$.\\
b. Deduce the value of the real number $u$ for which the distance $M N$ is minimal.