\textbf{(Candidates who did not follow the specialization course)}

We denote by $\mathbb { R }$ the set of real numbers. The space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). Consider the points $\mathrm { A } ( - 1 ; 2 ; 0 ) , \mathrm { B } ( 1 ; 2 ; 4 )$ and $\mathrm { C } ( - 1 ; 1 ; 1 )$.

\begin{enumerate}
  \item a. Prove that points $\mathrm { A } , \mathrm { B }$ and C are not collinear.\\
b. Calculate the dot product $\overrightarrow { \mathrm { AB } } \cdot \overrightarrow { \mathrm { AC } }$.\\
c. Deduce the measure of angle $\widehat { \mathrm { BAC } }$, rounded to the nearest degree.
  \item Let $\vec { n }$ be the vector with coordinates $\left( \begin{array} { c } 2 \\ - 1 \\ - 1 \end{array} \right)$.\\
a. Prove that $\vec { n }$ is a normal vector to plane (ABC).\\
b. Determine a Cartesian equation of plane ( ABC ).
  \item Let $\mathscr { P } _ { 1 }$ be the plane with equation $3 x + y - 2 z + 3 = 0$ and $\mathscr { P } _ { 2 }$ the plane passing through O and parallel to the plane with equation $x - 2 z + 6 = 0$.\\
a. Prove that plane $\mathscr { P } _ { 2 }$ has equation $x = 2z$.\\
b. Prove that planes $\mathscr{P}_1$, $\mathscr{P}_2$ and (ABC) have a common point, and determine its coordinates.
\end{enumerate}