We consider the rectangular prism ABCDEFGH such that $\mathrm{AB} = 3$ and $\mathrm{AD} = \mathrm{AE} = 1$.

We consider the point I on the segment $[\mathrm{AB}]$ such that $\overrightarrow{\mathrm{AB}} = 3\overrightarrow{\mathrm{AI}}$ and we call $M$ the midpoint of the segment [CD].\\
We place ourselves in the orthonormal coordinate system ($A$; $\overrightarrow{AI}, \overrightarrow{AD}, \overrightarrow{AE}$).

\begin{enumerate}
  \item Without justification, give the coordinates of the points $\mathrm{F}$, $\mathrm{H}$ and $M$.
  \item 
  \begin{enumerate}
    \item[a.] Show that the vector $\vec{n}\begin{pmatrix} 2 \\ 6 \\ 3 \end{pmatrix}$ is a normal vector to the plane (HMF).\\
    \item[b.] Deduce that a Cartesian equation of the plane (HMF) is:
    $$2x + 6y + 3z - 9 = 0$$
    \item[c.] Is the plane $\mathscr{P}$ whose Cartesian equation is $5x + 15y - 3z + 7 = 0$ parallel to the plane (HMF)? Justify your answer.
  \end{enumerate}
  \item Determine a parametric representation of the line (DG).
  \item We call $N$ the point of intersection of the line (DG) with the plane (HMF). Determine the coordinates of point N.
  \item Is the point R with coordinates $\left(3; \frac{1}{4}; \frac{1}{2}\right)$ the orthogonal projection of point G onto the plane (HMF)? Justify your answer.
\end{enumerate}