\textbf{Exercise 2}

Consider the cube ABCDEFGH.\\
We place the point M such that $\overrightarrow{\mathrm{BM}} = \overrightarrow{\mathrm{AB}}$.

\textbf{Part A}
\begin{enumerate}
  \item Show that the lines (FG) and (FM) are perpendicular.
  \item Show that the points A, M, G and H are coplanar.
\end{enumerate}

\textbf{Part B}

We place ourselves in the orthonormal coordinate system $(A;\overrightarrow{AB},\overrightarrow{AD},\overrightarrow{AE})$.

\begin{enumerate}
  \item Determine the coordinates of the vectors $\overrightarrow{\mathrm{GM}}$ and $\overrightarrow{\mathrm{AH}}$ and show that they are not collinear.
  \item \begin{enumerate}
    \item[a.] Justify that a parametric representation of the line (GM) is:
$$\left\{\begin{aligned} x &= 1+t \\ y &= 1-t \quad \text{with } t \in \mathbb{R}. \\ z &= 1-t \end{aligned}\right.$$
    \item[b.] We admit that a parametric representation of the line (AH) is:
$$\left\{\begin{aligned} x &= 0 \\ y &= k \\ z &= k \end{aligned}\right. \text{ with } k \in \mathbb{R}.$$
Show that the intersection point of (GM) and (AH), which we will call N, has coordinates $(0;2;2)$.
  \end{enumerate}
  \item \begin{enumerate}
    \item[a.] Show that the triangle AMN is a right-angled triangle at A.
    \item[b.] Calculate the area of this triangle.
  \end{enumerate}
  \item Let J be the centre of the face BCGF.
  \begin{enumerate}
    \item[a.] Determine the coordinates of point J.
    \item[b.] Show that the vector $\overrightarrow{\mathrm{FJ}}$ is a normal vector to the plane (AMN).
    \item[c.] Show that J belongs to the plane (AMN). Deduce that it is the orthogonal projection of point F onto the plane (AMN).
  \end{enumerate}
  \item We recall that the volume $V$ of a tetrahedron or a pyramid is given by the formula:
$$V = \frac{1}{3} \times \mathscr{B} \times h$$
$\mathscr{B}$ being the area of a base and $h$ the height relative to this base.\\
Show that the volume of the tetrahedron AMNF is twice the volume of the pyramid BCGFM.
\end{enumerate}