Exercise 2 (4 points) We consider a cube ABCDEFGH.
a. Simplify the vector $\overrightarrow{\mathrm{AC}} + \overrightarrow{\mathrm{AE}}$. b. Deduce that $\overrightarrow{\mathrm{AG}} \cdot \overrightarrow{\mathrm{BD}} = 0$. c. It is admitted that $\overrightarrow{\mathrm{AG}} \cdot \overrightarrow{\mathrm{BE}} = 0$. Prove that the line (AG) is orthogonal to the plane (BDE).
Space is equipped with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. a. Prove that a Cartesian equation of the plane (BDE) is $x + y + z - 1 = 0$. b. Determine the coordinates of the intersection point K of the line (AG) and the plane (BDE). c. It is admitted that the area, in square units, of triangle BDE is equal to $\dfrac{\sqrt{3}}{2}$. Calculate the volume of the pyramid BDEG.
\textbf{Exercise 2} (4 points)
We consider a cube ABCDEFGH.
\begin{enumerate}
\item a. Simplify the vector $\overrightarrow{\mathrm{AC}} + \overrightarrow{\mathrm{AE}}$.\\
b. Deduce that $\overrightarrow{\mathrm{AG}} \cdot \overrightarrow{\mathrm{BD}} = 0$.\\
c. It is admitted that $\overrightarrow{\mathrm{AG}} \cdot \overrightarrow{\mathrm{BE}} = 0$. Prove that the line (AG) is orthogonal to the plane (BDE).
\item Space is equipped with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.\\
a. Prove that a Cartesian equation of the plane (BDE) is $x + y + z - 1 = 0$.\\
b. Determine the coordinates of the intersection point K of the line (AG) and the plane (BDE).\\
c. It is admitted that the area, in square units, of triangle BDE is equal to $\dfrac{\sqrt{3}}{2}$. Calculate the volume of the pyramid BDEG.
\end{enumerate}