\textbf{Exercise 3 — 7 points}\\
Theme: Geometry in space\\
Space is equipped with an orthonormal coordinate system $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$.\\
We consider the points
$$\mathrm{A}(3;-2;2), \quad \mathrm{B}(6;1;5), \quad \mathrm{C}(6;-2;-1) \quad \text{and} \quad \mathrm{D}(0;4;-1).$$
We recall that the volume of a tetrahedron is given by the formula:
$$V = \frac{1}{3}\mathscr{A} \times h$$
where $\mathscr{A}$ is the area of the base and $h$ is the corresponding height.

\begin{enumerate}
  \item Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ are not coplanar.

  \item \begin{enumerate}
    \item[a.] Show that the triangle ABC is right-angled.
    \item[b.] Show that the line (AD) is perpendicular to the plane (ABC).
    \item[c.] Deduce the volume of the tetrahedron ABCD.
  \end{enumerate}

  \item We consider the point $\mathrm{H}(5;0;1)$.
  \begin{enumerate}
    \item[a.] Show that there exist real numbers $\alpha$ and $\beta$ such that $\overrightarrow{\mathrm{BH}} = \alpha\overrightarrow{\mathrm{BC}} + \beta\overrightarrow{\mathrm{BD}}$.
    \item[b.] Prove that H is the orthogonal projection of point A onto the plane (BCD).
    \item[c.] Deduce the distance from point A to the plane (BCD).
  \end{enumerate}

  \item Deduce from the previous questions the area of triangle BCD.
\end{enumerate}