We consider the cube ABCDEFGH represented below.\\
The points I and J are the midpoints of segments $[\mathrm{AB}]$ and $[\mathrm{CG}]$ respectively.\\
The point N is the midpoint of segment [IJ].\\
The objective of this exercise is to calculate the volume of the tetrahedron HFIJ.\\
We place ourselves in the orthonormal coordinate system ($A$; $\overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).

\begin{enumerate}
  \item a. Give the coordinates of points I and J.

Deduce the coordinates of N.\\
b. Justify that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ have the respective coordinates:
$$\overrightarrow{\mathrm{IJ}} \left(\begin{array}{c} 0.5 \\ 1 \\ 0.5 \end{array}\right) \text{ and } \overrightarrow{\mathrm{NF}} \left(\begin{array}{c} 0.25 \\ -0.5 \\ 0.75 \end{array}\right)$$
c. Prove that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ are orthogonal.

We admit that $\mathrm{NF} = \frac{\sqrt{14}}{4}$.\\
d. Deduce that the area of triangle FIJ is equal to $\frac{\sqrt{21}}{8}$.

  \item We consider the vector $\vec{u}\left(\begin{array}{c} 4 \\ -1 \\ -2 \end{array}\right)$.\\
a. Prove that the vector $\vec{u}$ is normal to the plane (FIJ).\\
b. Deduce that a Cartesian equation of the plane (FIJ) is: $4x - y - 2z - 2 = 0$.\\
c. We denote by $d$ the line perpendicular to the plane (FIJ) passing through point H. Determine a parametric representation of the line $d$.\\
d. Show that the distance from point H to the plane (FIJ) is equal to $\frac{5\sqrt{21}}{21}$.\\
e. We recall that the volume of a pyramid is given by the formula
$V = \frac{1}{3} \times \mathscr{B} \times h$ where $\mathscr{B}$ is the area of a base and $h$ is the length of the height relative to this base.\\
Calculate the volume of the tetrahedron HFIJ. Give the answer in the form of an irreducible fraction.
\end{enumerate}