A house consists of a rectangular parallelepiped ABCDEFGH topped with a prism EFIHGJ whose base is the triangle EIF isosceles at I.

We have $\mathrm { AB } = 3 , \quad \mathrm { AD } = 2 , \quad \mathrm { AE } = 1$.\\
We define the vectors $\vec { \imath } = \frac { 1 } { 3 } \overrightarrow { \mathrm { AB } } , \vec { \jmath } = \frac { 1 } { 2 } \overrightarrow { \mathrm { AD } } , \vec { k } = \overrightarrow { \mathrm { AE } }$.\\
We thus equip space with the orthonormal coordinate system $( \mathrm { A } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.

\begin{enumerate}
  \item Give the coordinates of point G.
  \item The vector $\vec { n }$ with coordinates $( 2 ; 0 ; - 3 )$ is a normal vector to the plane (EHI).

Determine a Cartesian equation of the plane (EHI).
  \item Determine the coordinates of point I.
  \item Determine a measure to the nearest degree of the angle $\widehat { \mathrm { EIF } }$.
  \item In order to connect the house to the electrical network, it is desired to dig a straight trench from an electrical relay located below the house.

The relay is represented by the point R with coordinates $( 6 ; - 3 ; - 1 )$.\\
The trench is assimilated to a segment of a line $\Delta$ passing through R and directed by the vector $\vec { u }$ with coordinates $( - 3 ; 4 ; 1 )$. It is desired to verify that the trench will reach the house at the level of the edge [BC].\\
a. Give a parametric representation of the line $\Delta$.\\
b. It is admitted that an equation of the plane (BFG) is $x = 3$.

Let K be the point of intersection of the line $\Delta$ with the plane (BFG).\\
Determine the coordinates of point K.\\
c. Does the point K indeed belong to the edge $[ \mathrm { BC } ]$?
\end{enumerate}