\section*{Exercise 3}
In the cube ABCDEFGH, we have placed the points $M$ and $N$ which are the midpoints of the segments $[ A B ]$ and $[ B C ]$ respectively.\\
We place ourselves in the coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$ ).
\begin{enumerate}
  \item Give without justification the coordinates of points $\mathrm { H } , \mathrm { M }$ and N.
  \item We admit that the lines (CD) and (MN) are secant and we denote K their point of intersection.\\
a. Give a parametric representation of the line (MN).\\
We admit that a parametric representation of the line (CD) is
$$\left\{ \begin{array} { l } 
x = t \\
y = 1 \\
z = 0
\end{array} , t \in \mathbb { R } . \right.$$
b. Determine the coordinates of point K.
  \item We admit that the points $\mathrm { H } , \mathrm { M } , \mathrm { N }$ define a plane and that the line (CG) and the plane (HMN) are secant. We denote L their point of intersection.\\
a. Verify that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 2 \\ 3 \end{array} \right)$ is a normal vector to the plane (HMN).\\
b. Determine a Cartesian equation of the plane (HMN).\\
c. Deduce the coordinates of point L.
  \item On ANNEX 2, construct the points K and L then the cross-section of the cube ABCDEFGH by the plane (HMN).
\end{enumerate}