Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).\\
Consider the points $\mathrm { A } ( 5 ; 5 ; 0 ) , \mathrm { B } ( 0 ; 5 ; 0 ) , \mathrm { C } ( 0 ; 0 ; 10 )$ and $\mathrm { D } \left( 0 ; 0 ; - \frac { 5 } { 2 } \right)$.

\begin{enumerate}
  \item a. Show that $\overrightarrow { n _ { 1 } } \left( \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right)$ is a normal vector to the plane (CAD).\\
b. Deduce that the plane (CAD) has the Cartesian equation: $x - y = 0$.
  \item Consider the line $\mathscr { D }$ with parametric representation $\left\{ \begin{aligned} x & = \frac { 5 } { 2 } t \\ y & = 5 - \frac { 5 } { 2 } t \text { where } t \in \mathbb { R } \text { . } \\ z & = 0 \end{aligned} \right.$\\
a. We admit that the line $\mathscr { D }$ and the plane (CAD) intersect at a point H. Justify that the coordinates of H are $\left( \frac { 5 } { 2 } ; \frac { 5 } { 2 } ; 0 \right)$.\\
b. Prove that the point H is the orthogonal projection of B onto the plane (CAD).
  \item a. Prove that the triangle ABH is right-angled at H.\\
b. Deduce that the area of triangle ABH is equal to $\frac { 25 } { 4 }$.
  \item a. Prove that ( CO ) is the height of the tetrahedron ABCH from C.\\
b. Deduce the volume of the tetrahedron ABCH.
\end{enumerate}

We recall that the volume of a tetrahedron is given by: $V = \frac { 1 } { 3 } \mathscr { B } h$, where $\mathscr { B }$ is the area of a base and h the height relative to this base.\\
5. We admit that the triangle ABC is right-angled at B. Deduce from the previous questions the distance from point H to the plane (ABC).