\textbf{Exercise 2}\\
Consider the cube ABCDEFGH with side length 1. The space is equipped with the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})$.
\begin{enumerate}
  \item a. Justify that the lines (AH) and (ED) are perpendicular.\\
b. Justify that the line (GH) is orthogonal to the plane (EDH).\\
c. Deduce that the line (ED) is orthogonal to the plane (AGH).
  \item Give the coordinates of the vector $\overrightarrow{\mathrm{ED}}$.\\
Deduce from question 1.c. that a Cartesian equation of the plane (AGH) is:
$$y - z = 0.$$
  \item Let L be the point with coordinates $\left(\frac{2}{3}; 1; 0\right)$.\\
a. Determine a parametric representation of the line (EL).\\
b. Determine the intersection of the line (EL) and the plane (AGH).\\
c. Prove that the orthogonal projection of point L onto the plane (AGH) is the point K with coordinates $\left(\frac{2}{3}; \frac{1}{2}; \frac{1}{2}\right)$.\\
d. Show that the distance from point L to the plane (AGH) is equal to $\frac{\sqrt{2}}{2}$.\\
e. Determine the volume of the tetrahedron LAGH.\\
Recall that the volume $V$ of a tetrahedron is given by the formula:
$$V = \frac{1}{3} \times (\text{area of the base}) \times \text{height}.$$
\end{enumerate}