We consider the cube ABCDEFGH, with edge length 1, represented below, and we equip space with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
Determine a parametric representation of the line (FD).
Prove that the vector $\vec{n}\begin{pmatrix}1\\-1\\1\end{pmatrix}$ is a normal vector to the plane (BGE) and determine an equation of the plane (BGE).
Show that the line (FD) is perpendicular to the plane (BGE) at a point K with coordinates $\mathrm{K}\left(\frac{2}{3}; \frac{1}{3}; \frac{2}{3}\right)$.
What is the nature of triangle BEG? Determine its area.
Deduce the volume of the tetrahedron BEGD.
\section*{Exercise 2 -- Common to all candidates}
We consider the cube ABCDEFGH, with edge length 1, represented below, and we equip space with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
\begin{enumerate}
\item Determine a parametric representation of the line (FD).
\item Prove that the vector $\vec{n}\begin{pmatrix}1\\-1\\1\end{pmatrix}$ is a normal vector to the plane (BGE) and determine an equation of the plane (BGE).
\item Show that the line (FD) is perpendicular to the plane (BGE) at a point K with coordinates $\mathrm{K}\left(\frac{2}{3}; \frac{1}{3}; \frac{2}{3}\right)$.
\item What is the nature of triangle BEG? Determine its area.
\item Deduce the volume of the tetrahedron BEGD.
\end{enumerate}