We consider two cubes ABCDEFGH and BKLCFJMG positioned as in the following figure. The point I is the midpoint of [EF]. We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}; \overrightarrow{AD}; \overrightarrow{AE})$. The points F, G and J have coordinates
$$\mathrm{F}(1;0;1), \quad \mathrm{G}(1;1;1) \quad \text{and} \quad \mathrm{J}(2;0;1).$$

\begin{enumerate}
  \item Show that the volume of the tetrahedron FIGB is equal to $\frac{1}{12}$ unit of volume.

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

  \item Determine the coordinates of point I.
  \item Show that the vector $\overrightarrow{\mathrm{DJ}}$ is a normal vector to the plane (BIG).
  \item Show that a Cartesian equation of the plane (BIG) is $2x - y + z - 2 = 0$.
  \item Determine a parametric representation of the line $d$, perpendicular to (BIG) and passing through F.
  \item a. The line $d$ intersects the plane (BIG) at point $\mathrm{L}'$. Show that the coordinates of point $\mathrm{L}'$ are $\left(\frac{2}{3}; \frac{1}{6}; \frac{5}{6}\right)$.\\
  b. Calculate the length $\mathrm{FL}'$.\\
  c. Deduce from the previous questions the area of triangle IGB.
\end{enumerate}