Exercise 4 -- For candidates who have not followed the specialized course
In space, consider the cube ABCDEFGH. We denote I and J the midpoints of segments [EH] and [FB] respectively. We equip space with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
Give the coordinates of points I and J.
a. Show that the vector $\vec{n}\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$ is a normal vector to the plane (BGI). b. Deduce a Cartesian equation of the plane (BGI). c. We denote K the midpoint of segment [HJ]. Does point K belong to the plane (BGI)?
The purpose of this question is to calculate the area of triangle BGI. a. Using for example triangle FIG as a base, prove that the volume of tetrahedron FBIG equals $\frac{1}{6}$. We recall that the volume $V$ of a tetrahedron is given by the formula $V = \frac{1}{3} \times \text{base area} \times \text{height}$.
\section*{Exercise 4 -- For candidates who have not followed the specialized course}
In space, consider the cube ABCDEFGH. We denote I and J the midpoints of segments [EH] and [FB] respectively.\\
We equip space with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
\begin{enumerate}
\item Give the coordinates of points I and J.
\item a. Show that the vector $\vec{n}\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$ is a normal vector to the plane (BGI).\\
b. Deduce a Cartesian equation of the plane (BGI).\\
c. We denote K the midpoint of segment [HJ]. Does point K belong to the plane (BGI)?
\item The purpose of this question is to calculate the area of triangle BGI.\\
a. Using for example triangle FIG as a base, prove that the volume of tetrahedron FBIG equals $\frac{1}{6}$.\\
We recall that the volume $V$ of a tetrahedron is given by the formula $V = \frac{1}{3} \times \text{base area} \times \text{height}$.
\end{enumerate}