In space, consider the cube ABCDEFGH with edge length equal to 1.\\
We equip the space with the orthonormal coordinate system (A ; $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$).\\
Consider the point M such that $\overrightarrow { \mathrm { BM } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { BH } }$.

\begin{enumerate}
  \item By reading the graph, give the coordinates of points $\mathrm { B } , \mathrm { D } , \mathrm { E } , \mathrm { G }$ and H.
  \item a. What is the nature of triangle EGD? Justify your answer.\\
b. It is admitted that the area of an equilateral triangle with side $c$ is equal to $\frac { \sqrt { 3 } } { 4 } c ^ { 2 }$.\\
Show that the area of triangle EGD is equal to $\frac { \sqrt { 3 } } { 2 }$.
  \item Prove that the coordinates of M are $\left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
  \item a. Justify that the vector $\vec { n } ( - 1 ; 1 ; 1 )$ is normal to the plane (EGD).\\
b. Deduce that a Cartesian equation of the plane (EGD) is: $- x + y + z - 1 = 0$.\\
c. Let $\mathscr { D }$ be the line perpendicular to the plane (EGD) and passing through point M. Show that a parametric representation of this line is:
$$\mathscr { D } : \left\{ \begin{aligned}
x & = \frac { 2 } { 3 } - t \\
y & = \frac { 1 } { 3 } + t , t \in \mathbb { R } \\
z & = \frac { 1 } { 3 } + t
\end{aligned} \right.$$
  \item The purpose of this question is to calculate the volume of the pyramid GEDM.\\
a. Let K be the foot of the height of the pyramid GEDM from point M.\\
Prove that the coordinates of point K are $\left( \frac { 1 } { 3 } ; \frac { 2 } { 3 } ; \frac { 2 } { 3 } \right)$.\\
b. Deduce the volume of the pyramid GEDM.\\
Recall that the volume $V$ of a pyramid is given by the formula\\
$V = \frac { b \times h } { 3 }$ where $b$ denotes the area of a base and h the associated height.
\end{enumerate}