Exercise 3 (7 points)\\
Theme: geometry in space\\
Consider a cube ABCDEFGH and call K the midpoint of segment [BC].\\
We place ourselves in the coordinate system $(A; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$ and consider the tetrahedron EFGK.\\
Recall that the volume of a tetrahedron is given by:
$$V = \frac{1}{3} \times \mathscr{B} \times h$$
where $\mathscr{B}$ denotes the area of a base and $h$ the height relative to this base.

\begin{enumerate}
  \item Specify the coordinates of points $\mathrm{E}, \mathrm{F}, \mathrm{G}$ and K.
  \item Show that the vector $\vec{n}\left(\begin{array}{r}2\\-2\\1\end{array}\right)$ is orthogonal to the plane (EGK).
  \item Prove that the plane (EGK) has the Cartesian equation: $2x - 2y + z - 1 = 0$.
  \item Determine a parametric representation of the line (d) orthogonal to the plane (EGK) passing through F.
  \item Show that the orthogonal projection $L$ of $F$ onto the plane (EGK) has coordinates $\left(\frac{5}{9}; \frac{4}{9}; \frac{7}{9}\right)$.
  \item Justify that the length LF is equal to $\frac{2}{3}$.
  \item Calculate the area of triangle EFG. Deduce that the volume of tetrahedron EFGK is equal to $\frac{1}{6}$.
  \item Deduce from the previous questions the area of triangle EGK.
  \item Consider the points P midpoint of segment [EG], M midpoint of segment [EK] and N midpoint of segment [GK]. Determine the volume of tetrahedron FPMN.
\end{enumerate}