\section*{Exercise 4}
We place ourselves in an orthonormal frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space.\\
We consider the points $\mathrm{A}(1; 0; 3)$, $\mathrm{B}(-2; 1; 2)$ and $\mathrm{C}(0; 3; 2)$.

\begin{enumerate}
  \item a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.\\
b. Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 4 \end{array}\right)$. Verify that the vector $\vec{n}$ is orthogonal to the plane (ABC).\\
c. Deduce that the plane $(\mathrm{ABC})$ has for Cartesian equation $-x + y + 4z - 11 = 0$.
\end{enumerate}

We consider the plane $\mathscr{P}$ with Cartesian equation $3x - 3y + 2z - 9 = 0$ and the plane $\mathscr{P}'$ with Cartesian equation $x - y - z + 2 = 0$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item a. Prove that the planes $\mathscr{P}$ and $\mathscr{P}'$ are secant. We denote by (d) their line of intersection.\\
b. Determine whether the planes $\mathscr{P}$ and $\mathscr{P}'$ are perpendicular.
  \item Show that the line (d) is directed by the vector $\vec{u}\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right)$.
  \item Show that the point $\mathrm{M}(2; 1; 3)$ belongs to the planes $\mathscr{P}$ and $\mathscr{P}'$. Deduce a parametric representation of the line (d).
  \item Show that the line (d) is also included in the plane (ABC). What can we say about the three planes (ABC), $\mathscr{P}$ and $\mathscr{P}'$?
\end{enumerate}