In space equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ) with unit 1 cm, we consider the following points:

$$\mathrm { J } ( 2 ; 0 ; 1 ) , \quad \mathrm { K } ( 1 ; 2 ; 1 ) \text { and } \quad \mathrm { L } ( - 2 ; - 2 ; - 2 )$$

\begin{enumerate}
  \item a. Show that triangle JKL is right-angled at J.\\
b. Calculate the exact value of the area of triangle JKL in $\mathrm { cm } ^ { 2 }$.\\
c. Determine an approximate value to the nearest tenth of the geometric angle $\widehat { \mathrm { JKL } }$.

  \item a. Prove that the vector $\vec { n }$ with coordinates $\left( \begin{array} { c } 6 \\ 3 \\ - 10 \end{array} \right)$ is a normal vector to the plane (JKL).\\
b. Deduce a Cartesian equation of the plane (JKL).
\end{enumerate}

In the following, T denotes the point with coordinates ( $10 ; 9 ; - 6$ ).\\
3. a. Determine a parametric representation of the line $\Delta$ perpendicular to the plane (JKL) and passing through T.\\
b. Determine the coordinates of point H, the orthogonal projection of point T onto the plane (JKL).\\
c. We recall that the volume $V$ of a tetrahedron is given by the formula:

$$V = \frac { 1 } { 3 } \mathscr { B } \times h \text { where } \mathscr { B } \text { denotes the area of a base and } h \text { the corresponding height }$$

Calculate the exact value of the volume of tetrahedron JKLT in $\mathrm { cm } ^ { 3 }$.