Consider the right prism ABFEDCGH, with base ABFE, a right trapezoid at A.\\
We associate with this prism the orthonormal coordinate system $(A; \vec{\imath}, \vec{\jmath}, \vec{k})$ such that:

$$\vec{\imath} = \frac{1}{4}\overrightarrow{AB}, \quad \vec{\jmath} = \frac{1}{4}\overrightarrow{AD}, \quad \vec{k} = \frac{1}{8}\overrightarrow{AE}$$

Moreover we have $\overrightarrow{BF} = \frac{1}{2}\overrightarrow{AE}$.\\
We denote I the midpoint of segment $[EF]$.\\
We denote J the midpoint of segment $[AE]$.

\begin{enumerate}
  \item Give the coordinates of points I and J.
  \item Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 1 \end{array}\right)$.\\
a. Show that the vector $\vec{n}$ is normal to the plane (IGJ).\\
b. Determine a Cartesian equation of the plane (IGJ).
  \item Determine a parametric representation of the line $d$, perpendicular to the plane (IGJ) and passing through H.
  \item We denote L the orthogonal projection of point H onto the plane (IGJ). Show that the coordinates of L are $\left(\frac{8}{3}; \frac{4}{3}; \frac{16}{3}\right)$.
  \item Calculate the distance from point H to the plane (IGJ).
  \item Show that triangle IGJ is right-angled at I.
  \item Deduce the volume of tetrahedron IGJH.
\end{enumerate}

We recall that the volume $V$ of a tetrahedron is given by the formula:
$$V = \frac{1}{3} \times \text{(area of base) \times height.}$$