Space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider:
\begin{itemize}
  \item the points $\mathrm{A}(-2; 0; 2)$, $\mathrm{B}(-1; 3; 0)$, $\mathrm{C}(1; -1; 2)$ and $\mathrm{D}(0; 0; 3)$.
  \item the line $\mathscr{D}_1$ whose parametric representation is $\left\{ \begin{aligned} x &= t \\ y &= 3t \\ z &= 3 + 5t \end{aligned} \right.$ with $t \in \mathbb{R}$.
  \item the line $\mathscr{D}_2$ whose parametric representation is $\left\{ \begin{aligned} x &= 1 + 3s \\ y &= -1 - 5s \\ z &= 2 - 6s \end{aligned} \right.$ with $s \in \mathbb{R}$.
\end{itemize}

\begin{enumerate}
  \item Prove that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
  \item \begin{enumerate}
    \item[a.] Prove that the vector $\vec{n}\begin{pmatrix} 1 \\ 3 \\ 5 \end{pmatrix}$ is orthogonal to the plane (ABC).
    \item[b.] Justify that a Cartesian equation of the plane (ABC) is:
$$x + 3y + 5z - 8 = 0$$
    \item[c.] Deduce that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ are not coplanar.
  \end{enumerate}
  \item \begin{enumerate}
    \item[a.] Justify that the line $\mathscr{D}_1$ is the altitude of the tetrahedron ABCD from D. It is admitted that the line $\mathscr{D}_2$ is the altitude of the tetrahedron ABCD from C.
    \item[b.] Prove that the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant and determine the coordinates of their point of intersection.
  \end{enumerate}
  \item \begin{enumerate}
    \item[a.] Determine the coordinates of the orthogonal projection H of point D onto the plane (ABC).
    \item[b.] Calculate the distance from point D to the plane (ABC). Round the result to the nearest hundredth.
  \end{enumerate}
\end{enumerate}