Exercise 2 — 5 points\\
Theme: geometry in space\\
Space is equipped with an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
We consider:
\begin{itemize}
  \item $d_1$ the line passing through point $H(2; 3; 0)$ with direction vector $\vec{u}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$;
  \item $d_2$ the line with parametric representation:
\end{itemize}
$$\left\{\begin{aligned}
x &= 2k - 3\\
y &= k\\
z &= 5
\end{aligned}\quad\text{where }k\text{ describes }\mathbb{R}.\right.$$
The purpose of this exercise is to determine a parametric representation of a line $\Delta$ that is perpendicular to both lines $d_1$ and $d_2$.
\begin{enumerate}
  \item a. Determine a direction vector $\vec{v}$ of line $d_2$.\\
b. Prove that lines $d_1$ and $d_2$ are not parallel.\\
c. Prove that lines $d_1$ and $d_2$ are not intersecting.\\
d. What is the relative position of lines $d_1$ and $d_2$?
  \item a. Verify that the vector $\vec{w}\left(\begin{array}{c}-1\\2\\3\end{array}\right)$ is orthogonal to both $\vec{u}$ and $\vec{v}$.\\
b. We consider the plane $P$ passing through point $H$ and directed by vectors $\vec{u}$ and $\vec{w}$. We admit that a Cartesian equation of this plane is:
$$5x + 4y - z - 22 = 0.$$
Prove that the intersection of plane $P$ and line $d_2$ is the point $M(3; 3; 5)$.
  \item Let $\Delta$ be the line with direction vector $\vec{w}$ passing through point $M$.\\
A parametric representation of $\Delta$ is therefore given by:
$$\left\{\begin{array}{l}
x = -r + 3\\
y = 2r + 3\\
z = 3r + 5
\end{array}\text{ where }r\text{ describes }\mathbb{R}.\right.$$
a. Justify that lines $\Delta$ and $d_1$ are perpendicular at a point $L$ whose coordinates you will determine.\\
b. Explain why line $\Delta$ is a solution to the problem posed.
\end{enumerate}