The cube ABCDEFGH has edge length 1 cm. The point I is the midpoint of segment [AB] and the point J is the midpoint of segment [CG].

We place ourselves in 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 Show that the vector $\overrightarrow{\mathrm{EJ}}$ is normal to the plane (FHI).
  \item Show that a Cartesian equation of the plane (FHI) is $-2x - 2y + z + 1 = 0$.
  \item Determine a parametric representation of the line (EJ).
  \item \begin{enumerate}
    \item[a.] We denote K the orthogonal projection of point E onto the plane $(\mathrm{FHI})$. Calculate its coordinates.
    \item[b.] Show that the volume of the pyramid EFHI is $\frac{1}{6}\mathrm{~cm}^3$.
  \end{enumerate}
  We may use the point L, midpoint of segment $[\mathrm{EF}]$. We admit that this point is the orthogonal projection of point I onto the plane (EFH).
  \begin{enumerate}
    \item[c.] Deduce from the two previous questions the area of triangle FHI.
  \end{enumerate}
\end{enumerate}