\textbf{Exercise 3 (7 points)}\\
Theme: geometry in space\\
An exhibition of contemporary art takes place in a room in the shape of a rectangular parallelepiped with width 6 m, length 8 m and height 4 m.\\
It is represented by the rectangular parallelepiped OBCDEFGH where $\mathrm { OB } = 6 \mathrm {~m} , \mathrm { OD } = 8 \mathrm {~m}$ and $\mathrm { OE } = 4 \mathrm {~m}$. We use the orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$ such that $\vec { \imath } = \frac { 1 } { 6 } \overrightarrow { \mathrm { OB } } , \vec { \jmath } = \frac { 1 } { 8 } \overrightarrow { \mathrm { OD } }$ and $\vec { k } = \frac { 1 } { 4 } \overrightarrow { \mathrm { OE } }$.\\
In this coordinate system we have, in particular $\mathrm { C } ( 6 ; 8 ; 0 ) , \mathrm { F } ( 6 ; 0 ; 4 )$ and $\mathrm { G } ( 6 ; 8 ; 4 )$.\\
One of the exhibited works is a glass triangle represented by triangle ART which has vertices $\mathrm { A } ( 6 ; 0 ; 2 )$, $\mathrm { R } ( 6 ; 3 ; 4 )$ and $\mathrm { T } ( 3 ; 0 ; 4 )$. Finally, S is the point with coordinates $\left( 3 ; \frac { 5 } { 2 } ; 0 \right)$.

\begin{enumerate}
  \item a. Verify that triangle ART is isosceles with apex A.\\
b. Calculate the dot product $\overrightarrow { \mathrm { AR } } \cdot \overrightarrow { \mathrm { AT } }$.\\
c. Deduce an approximate value to 0.1 degree of the angle $\widehat { \mathrm { RAT } }$.
  \item a. Justify that the vector $\vec { n } \left( \begin{array} { c } 2 \\ - 2 \\ 3 \end{array} \right)$ is a normal vector to the plane (ART).\\
b. Deduce a Cartesian equation of the plane (ART).
  \item A laser beam directed towards triangle ART is emitted from the floor from point S. It is admitted that this beam is perpendicular to the plane (ART).\\
a. Let $\Delta$ be the line perpendicular to the plane (ART) and passing through point S. Justify that the system below is a parametric representation of the line $\Delta$:
$$\left\{ \begin{aligned}
x & = 3 + 2 k \\
y & = \frac { 5 } { 2 } - 2 k , \text { with } k \in \mathbb { R } . \\
z & = 3 k
\end{aligned} \right.$$
b. Let L be the point of intersection of the line $\Delta$ with the plane (ART). Prove that L has coordinates $\left( 5 ; \frac { 1 } { 2 } ; 3 \right)$.
  \item The artist installs a rail represented by the segment [DK] where K is the midpoint of segment [EH]. On this rail, he positions a laser light source at a point N of segment [DK] and directs this second laser beam towards point S.\\
a. Show that, for every real $t$ in the interval $[ 0 ; 1 ]$, the point N with coordinates $( 0 ; 8 - 4 t ; 4 t )$ is a point of segment [DK].\\
b. Calculate the exact coordinates of point N such that the two laser beams represented by segments [SL] and [SN] are perpendicular.
\end{enumerate}