Consider a cube ABCDEFGH and the space is referred to the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.\\
For any real $m$ belonging to the interval $[0; 1]$, we consider the points $K$ and $L$ with coordinates:
$$K(m; 0; 0) \text{ and } L(1-m; 1; 1).$$
\begin{enumerate}
  \item Give the coordinates of points E and C in this coordinate system.
  \item In this question, $m = 0$. Thus, the point $\mathrm{L}(1; 1; 1)$ coincides with point G, the point $\mathrm{K}(0; 0; 0)$ coincides with point A and the plane (LEK) is therefore the plane (GEA).\\
a. Justify that the vector $\overrightarrow{\mathrm{DB}} \left(\begin{array}{c} 1 \\ -1 \\ 0 \end{array}\right)$ is normal to the plane (GEA).\\
b. Determine a Cartesian equation of the plane (GEA).
  \item In this question, $m$ is any real number in the interval $[0; 1]$.\\
a. Prove that $\mathrm{CKEL}$ is a parallelogram.\\
b. Justify that $\overrightarrow{KC} \cdot \overrightarrow{KE} = m(m-1)$.\\
c. Prove that $\mathrm{CKEL}$ is a rectangle if, and only if, $m = 0$ or $m = 1$.
  \item In this question, $m = \frac{1}{2}$. Thus, L has coordinates $\left(\frac{1}{2}; 1; 1\right)$ and K has coordinates $\left(\frac{1}{2}; 0; 0\right)$.\\
a. Prove that the parallelogram CKEL is then a rhombus.\\
b. Using question 3.b., determine an approximate value to the nearest degree of the measure of the angle $\widehat{\mathrm{CKE}}$.
\end{enumerate}