An artist wishes to create a sculpture composed of a tetrahedron placed on a cube with 6-metre edges. These two solids are represented by the cube $ABCDEFGH$ and by the tetrahedron $SELM$.

The space is equipped with an orthonormal coordinate system $(A; \overrightarrow{AI}, \overrightarrow{AJ}, \overrightarrow{AK})$ such that: $I \in [AB]$, $J \in [AD]$, $K \in [AE]$ and $AI = AJ = AK = 1$, the graphical unit representing 1 metre.

The points $L$, $M$ and $S$ are defined as follows:
\begin{itemize}
  \item $L$ is the point such that $\overrightarrow{FL} = \frac{2}{3}\overrightarrow{FE}$;
  \item $M$ is the point of intersection of the plane $(BDL)$ and the line $(EH)$;
  \item $S$ is the point of intersection of the lines $(BL)$ and $(AK)$.
\end{itemize}

\begin{enumerate}
  \item Prove, without calculating coordinates, that the lines $(LM)$ and $(BD)$ are parallel.
  \item Prove that the coordinates of point $L$ are $(2; 0; 6)$.
  \item a. Give a parametric representation of the line $(BL)$.\\
  b. Verify that the coordinates of point $S$ are $(0; 0; 9)$.
  \item Let $\vec{n}$ be the vector with coordinates $(3; 3; 2)$.\\
  a. Verify that $\vec{n}$ is normal to the plane $(BDL)$.\\
  b. Prove that a Cartesian equation of the plane $(BDL)$ is:
  $$3x + 3y + 2z - 18 = 0$$
  c. It is admitted that the line $(EH)$ has the parametric representation:
  $$\left\{\begin{array}{l} x = 0 \\ y = s \\ z = 6 \end{array} \quad (s \in \mathbb{R})\right.$$
  Calculate the coordinates of point $M$.
  \item Calculate the volume of the tetrahedron $SELM$. Recall that the volume $V$ of a tetrahedron is given by the following formula:
  $$V = \frac{1}{3} \times \text{Area of base} \times \text{Height}$$
  \item The artist wishes the measure of angle $\widehat{SLE}$ to be between $55^\circ$ and $60^\circ$. Is this angle constraint satisfied?
\end{enumerate}