We consider the cube ABCDEFGH which is represented in APPENDIX.\\
In the orthonormal coordinate system ( $A$; $\overrightarrow { A B }$; $\overrightarrow { A D }$; $\overrightarrow { A E }$ ), we consider the points $M , N$ and $P$ with coordinates:

$$\mathrm { M } \left( 1 ; 1 ; \frac { 3 } { 4 } \right) , \quad \mathrm { N } \left( 0 ; \frac { 1 } { 2 } ; 1 \right) , \quad \mathrm { P } \left( 1 ; 0 ; - \frac { 5 } { 4 } \right)$$

In this exercise, we propose to calculate the volume of the tetrahedron FMNP.

\begin{enumerate}
  \item Give the coordinates of the vectors $\overrightarrow { \mathrm { MN } }$ and $\overrightarrow { \mathrm { MP } }$.
  \item Place the points $\mathrm { M} , \mathrm { N }$ and P on the figure given in APPENDIX which must be returned with your work.
  \item Justify that the points $\mathrm { M } , \mathrm { N }$ and P are not collinear.
\end{enumerate}

From then on, the three points define the plane (MNP).\\
4. a. Calculate the dot product $\overrightarrow { \mathrm { MN } } \cdot \overrightarrow { \mathrm { MP } }$, then deduce the nature of the triangle MNP.\\
b. Calculate the area of the triangle MNP.\\
5. a. Show that the vector $\vec { n } ( 5 ; - 8 ; 4 )$ is a normal vector to the plane (MNP).\\
b. Deduce that a Cartesian equation of the plane (MNP) is $5 x - 8 y + 4 z = 0$.\\
6. We recall that the point F has coordinates $\mathrm { F } ( 1 ; 0 ; 1 )$.

Determine a parametric representation of the line $d$ orthogonal to the plane (MNP) and passing through the point F.\\
7. We denote L the orthogonal projection of the point F onto the plane (MNP).

Show that the coordinates of the point L are: $\mathrm { L } \left( \frac { 4 } { 7 } ; \frac { 24 } { 35 } ; \frac { 23 } { 35 } \right)$.\\
8. Show that $\mathrm { FL } = \frac { 3 \sqrt { 105 } } { 35 }$ then calculate the volume of the tetrahedron FMNP.

We recall that the volume V of a tetrahedron is given by the formula:

$$V = \frac { 1 } { 3 } \times \text { area of a base } \times \text{ height associated with this base. }$$