In space equipped with an orthonormal coordinate system with unit 1 cm, we consider the points

$$\mathrm{D}(3;1;5), \quad \mathrm{E}(3;-2;-1), \quad \mathrm{F}(-1;2;1), \quad \mathrm{G}(3;2;-3).$$

\begin{enumerate}
  \item a. Determine the coordinates of the vectors $\overrightarrow{\mathrm{EF}}$ and $\overrightarrow{\mathrm{FG}}$.\\
b. Justify that the points $\mathrm{E}$, $\mathrm{F}$ and $\mathrm{G}$ are not collinear.
  \item a. Determine a parametric representation of the line (FG).\\
b. We call H the point with coordinates $(2; 2; -2)$. Verify that H is the orthogonal projection of E onto the line (FG).\\
c. Show that the area of triangle EFG is equal to $12\text{ cm}^{2}$.
  \item a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (EFG).\\
b. Determine a Cartesian equation of the plane (EFG).\\
c. Determine a parametric representation of the line $(d)$ passing through point D and orthogonal to the plane (EFG).\\
d. We denote K the orthogonal projection of point D onto the plane (EFG). Using the previous questions, calculate the coordinates of point K.
  \item a. Verify that the distance $DK$ is equal to 5 cm.\\
b. Deduce the volume of the tetrahedron DEFG.
\end{enumerate}