\section*{Exercise 3 (5 points)}

We place ourselves in space equipped with an orthonormal coordinate system whose origin is point A.\\
We consider the points $\mathrm{B}(10; -8; 2)$, $\mathrm{C}(-1; -8; 5)$ and $\mathrm{D}(14; 4; 8)$.

\begin{enumerate}
  \item a. Determine a system of parametric equations for each of the lines (AB) and (CD).\\
b. Verify that the lines (AB) and (CD) are not coplanar.
  \item We consider the point I on the line (AB) with abscissa 5 and the point J on the line (CD) with abscissa 4.\\
a. Determine the coordinates of points I and J and deduce the distance IJ.\\
b. Demonstrate that the line (IJ) is perpendicular to the lines (AB) and (CD). The line (IJ) is called the common perpendicular to the lines (AB) and (CD).
  \item The purpose of this question is to verify that the distance IJ is the minimum distance between the lines (AB) and (CD).\\
We consider a point $M$ on the line (AB) distinct from point I.\\
We consider a point $M'$ on the line (CD) distinct from point J.\\
a. Justify that the parallel to the line (IJ) passing through point $M'$ intersects the line $\Delta$ (the line parallel to (CD) passing through I) at a point that we will denote $P$.\\
b. Demonstrate that the triangle $MPM'$ is right-angled at $P$.\\
c. Justify that $MM' > IJ$ and conclude.
\end{enumerate}