Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
Consider the points
$$\mathrm{A}(1;0;-1), \quad \mathrm{B}(3;-1;2), \quad \mathrm{C}(2;-2;-1) \quad \text{and} \quad \mathrm{D}(4;-1;-2).$$
We denote by $\Delta$ the line with parametric representation
$$\left\{\begin{aligned} x &= 0 \\ y &= 2+t, \text{ with } t \in \mathbb{R}. \\ z &= -1+t \end{aligned}\right.$$

\begin{enumerate}
  \item a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ define a plane which we will denote $\mathscr{P}$.\\
  b. Show that the line (CD) is orthogonal to the plane $\mathscr{P}$. On the plane $\mathscr{P}$, what does point C represent with respect to D?\\
  c. Show that a Cartesian equation of the plane $\mathscr{P}$ is: $2x + y - z - 3 = 0$.
  \item a. Calculate the distance CD.\\
  b. Does there exist a point M on the plane $\mathscr{P}$ different from C satisfying $\mathrm{MD} = \sqrt{6}$? Justify your answer.
  \item a. Show that the line $\Delta$ is contained in the plane $\mathscr{P}$.\\
  Let H be the orthogonal projection of point D onto the line $\Delta$.\\
  b. Show that H is the point of $\Delta$ associated with the value $t = -2$ in the parametric representation of $\Delta$ given above.\\
  c. Deduce the distance from point D to the line $\Delta$.
\end{enumerate}