\textbf{Exercise 2 — 7 points}

Theme: Geometry in space

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

\textbf{1. Calculation of an angle}

a. Calculate the coordinates of the vectors $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$ and deduce that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.\\
b. Calculate the lengths AB and AC.\\
c. Using the dot product $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}$, determine the value of the cosine of the angle $\widehat{\mathrm{BAC}}$ then give an approximate value of the measure of the angle $\widehat{\mathrm{BAC}}$ to the nearest tenth of a degree.

\textbf{2. Calculation of an area}

a. Determine an equation of the plane $\mathscr{P}$ passing through point C and perpendicular to the line (AB).\\
b. Give a parametric representation of the line (AB).\\
c. Deduce the coordinates of the orthogonal projection E of point C onto the line $(\mathrm{AB})$, that is to say the point of intersection of the line (AB) and the plane $\mathscr{P}$.\\
d. Calculate the area of triangle ABC.

\textbf{3. Calculation of a volume}

a. Let the point $\mathrm{F}(1; -1; 3)$. Show that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{F}$ are coplanar.\\
b. Verify that the line (FD) is orthogonal to the plane (ABC).\\
c. Knowing that the volume of a tetrahedron is equal to one third of the area of its base multiplied by its height, calculate the volume of the tetrahedron ABCD.