We consider the cube ABCDEFGH with edge 1. We call I the point of intersection of the plane (GBD) with the line (EC). The space is referred to the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.

\begin{enumerate}
  \item Give in this coordinate system the coordinates of points $\mathrm{E}, \mathrm{C}, \mathrm{G}$.
  \item Determine a parametric representation of the line (EC).
  \item Prove that the line (EC) is orthogonal to the plane (GBD).
  \item \begin{enumerate}
    \item[a.] Justify that a Cartesian equation of the plane (GBD) is:
$$x + y - z - 1 = 0.$$
    \item[b.] Show that point I has coordinates $\left(\frac{2}{3}; \frac{2}{3}; \frac{1}{3}\right)$.
    \item[c.] Deduce that the distance from point E to the plane (GBD) is equal to $\frac{2\sqrt{3}}{3}$.
  \end{enumerate}
  \item \begin{enumerate}
    \item[a.] Prove that triangle BDG is equilateral.
    \item[b.] Calculate the area of triangle BDG. You may use point J, the midpoint of segment [BD].
  \end{enumerate}
  \item Justify that the volume of tetrahedron EGBD is equal to $\frac{1}{3}$.

We recall that the volume of a tetrahedron is given by $V = \frac{1}{3}Bh$ where $B$ is the area of a base of the tetrahedron and $h$ is the height relative to this base.
\end{enumerate}