In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points:

$$A(4; -4; 4), \quad B(5; -3; 2), \quad C(6; -2; 3), \quad D(5; 1; 1)$$

\begin{enumerate}
  \item Prove that triangle ABC is right-angled at $B$.
  \item Justify that a Cartesian equation of the plane (ABC) is:
$$x - y - 8 = 0.$$
  \item We denote $d$ the line passing through point $D$ and perpendicular to the plane (ABC).\\
  a. Determine a parametric representation of the line $d$.\\
  b. We denote H the orthogonal projection of point $D$ onto the plane $(ABC)$. Determine the coordinates of point H.\\
  c. Show that $DH = 2\sqrt{2}$.
  \item a. Show that the volume of the pyramid ABCD is equal to 2. We recall that the volume V of a pyramid is calculated using the formula:
$$V = \frac{1}{3} \times \mathscr{B} \times h$$
where $\mathscr{B}$ is the area of a base of the pyramid and $h$ is the corresponding height.\\
  b. We admit that the area of triangle BCD is equal to $\frac{\sqrt{42}}{2}$. Deduce the exact value of the distance from point A to the plane (BCD).
\end{enumerate}