Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).\\
Consider the points

$$\mathrm { A } ( 3 ; 0 ; 1 ) , \quad \mathrm { B } ( 2 ; 1 ; 2 ) \text { and } \quad \mathrm { C } ( - 2 ; - 5 ; 1 ) .$$

\begin{enumerate}
  \item Prove that the points $\mathrm { A } , \mathrm { B }$ and C are not collinear.
  \item Prove that the triangle ABC is right-angled at A .
  \item Verify that the plane $( \mathrm { ABC } )$ has the Cartesian equation :
\end{enumerate}

$$- x + y - 2 z + 5 = 0$$

\begin{enumerate}
  \setcounter{enumi}{3}
  \item Consider the point $S ( 1 ; - 2 ; 4 )$.
\end{enumerate}

Determine the parametric representation of the line ( $\Delta$ ), passing through S and orthogonal to the plane (ABC).\\
5. We call H the point of intersection of the line ( $\Delta$ ) and the plane (ABC).

Show that the coordinates of H are $( 0 ; - 1 ; 2 )$.\\
6. Calculate the exact value of the distance SH.\\
7. Consider the circle $\mathscr { C }$, included in the plane (ABC), with center H, passing through the point B. We call $\mathscr { D }$ the disk bounded by the circle $\mathscr { C }$.

Determine the exact value of the area of the disk $\mathscr { D }$.\\
8. Deduce the exact value of the volume of the cone with apex S and base the disk $\mathscr { D }$.