\textbf{Exercise 4: Geometry in Space}

In space equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points
$$\mathrm{A}(0; 8; 6), \quad \mathrm{B}(6; 4; 4) \quad \text{and} \quad \mathrm{C}(2; 4; 0).$$
\begin{enumerate}
  \item a. Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.\\
    b. Show that the vector $\vec{n}(1; 2; -1)$ is a normal vector to the plane (ABC).\\
    c. Determine a Cartesian equation of the plane (ABC).
  \item Let D and E be the points with respective coordinates $(0; 0; 6)$ and $(6; 6; 0)$.\\
    a. Determine a parametric representation of the line (DE).\\
    b. Show that the midpoint I of the segment [BC] belongs to the line (DE).
  \item We consider the triangle ABC.\\
    a. Determine the nature of triangle ABC.\\
    b. Calculate the area of triangle ABC in square units.\\
    c. Calculate $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}$.\\
    d. Deduce a measure of the angle $\widehat{\mathrm{BAC}}$ rounded to 0.1 degree.
  \item We consider the point H with coordinates $\left(\dfrac{5}{3}; \dfrac{10}{3}; -\dfrac{5}{3}\right)$.\\
    Show that $H$ is the orthogonal projection of the point $O$ onto the plane (ABC).\\
    Deduce the distance from point O to the plane (ABC).
\end{enumerate}