Consider a rectangular parallelepiped ABCDEFGH such that $\mathrm{AB} = \mathrm{AD} = 1$ and $\mathrm{AE} = 2$. Point I is the midpoint of segment [AE]. Point K is the midpoint of segment [DC]. Point L is defined by: $\overrightarrow{\mathrm{DL}} = \frac{3}{2}\overrightarrow{\mathrm{AI}}$. N is the orthogonal projection of point D onto the plane (AKL).

We use the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AI}})$.\\
We admit that point L has coordinates $\left(0; 1; \frac{3}{2}\right)$.

\begin{enumerate}
  \item Determine the coordinates of vectors $\overrightarrow{\mathrm{AK}}$ and $\overrightarrow{\mathrm{AL}}$.
  \item a. Prove that the vector $\vec{n}$ with coordinates $(6; -3; 2)$ is a normal vector to the plane (AKL).\\
b. Deduce a Cartesian equation of the plane (AKL).\\
c. Determine a system of parametric equations of the line $\Delta$ passing through D and perpendicular to the plane (AKL).\\
d. Deduce that the point N with coordinates $\left(\frac{18}{49}; \frac{40}{49}; \frac{6}{49}\right)$ is the orthogonal projection of point D onto the plane (AKL).
\end{enumerate}

We recall that the volume $\mathcal{V}$ of a tetrahedron is given by the formula:
$$\mathcal{V} = \frac{1}{3} \times (\text{area of the base}) \times \text{height.}$$

\begin{enumerate}
  \setcounter{enumi}{2}
  \item a. Calculate the volume of tetrahedron ADKL using triangle ADK as the base.\\
b. Calculate the distance from point D to the plane (AKL).\\
c. Deduce from the previous questions the area of triangle AKL.
\end{enumerate}