In space with respect to an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points: A with coordinates $(2; 0; 0)$, B with coordinates $(0; 3; 0)$ and C with coordinates $(0; 0; 1)$.

The objective of this exercise is to calculate the area of triangle ABC.

\begin{enumerate}
  \item a. Show that the vector $\vec{n}\left(\begin{array}{l}3\\2\\6\end{array}\right)$ is normal to the plane (ABC).\\
b. Deduce that a Cartesian equation of the plane (ABC) is: $3x + 2y + 6z - 6 = 0$.
  \item We denote by $d$ the line passing through O and perpendicular to the plane (ABC).\\
a. Determine a parametric representation of the line $d$.\\
b. Show that the line $d$ intersects the plane (ABC) at the point H with coordinates $\left(\frac{18}{49}; \frac{12}{49}; \frac{36}{49}\right)$.\\
c. Calculate the distance OH.
  \item We recall that the volume of a pyramid is given by: $V = \frac{1}{3}\mathscr{B}h$, where $\mathscr{B}$ is the area of a base and $h$ is the height of the pyramid corresponding to this base.\\
By calculating in two different ways the volume of the pyramid OABC, determine the area of triangle ABC.
\end{enumerate}