A municipality decides to replace the traditional July 14 fireworks with a luminous drone show. For drone piloting, space is equipped with an orthonormal reference frame $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$ whose unit is one hundred meters. The position of each drone is modeled by a point and each drone is sent from a starting point D with coordinates $(2;5;1)$. It is desired to form figures with drones by positioning them in the same plane $\mathscr{P}$. Three drones are positioned at points $\mathrm{A}(-1;-1;17)$, $\mathrm{B}(4;-2;4)$ and $\mathrm{C}(1;-3;7)$.
Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. In the following, we denote by $\mathscr{P}$ the plane (ABC) and we consider the vector $\vec{n}\begin{pmatrix}2\\-3\\1\end{pmatrix}$.
a. Justify that $\vec{n}$ is normal to the plane (ABC). b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $2x - 3y + z - 18 = 0$.
The drone pilot decides to send a fourth drone taking as trajectory the line $d$ whose parametric representation is given by $$d : \left\{\begin{array}{rl} x &= 3t + 2 \\ y &= t + 5 \\ z &= 4t + 1 \end{array},\text{ with } t \in \mathbb{R}.\right.$$ a. Determine a direction vector of the line $d$. b. So that this new drone is also placed in the plane $\mathscr{P}$, determine by calculation the coordinates of point E, the intersection of the line $d$ with the plane $\mathscr{P}$.
The drone pilot decides to send a fifth drone along the line $\Delta$ which passes through point $\mathrm{D}$ and which is perpendicular to the plane $\mathscr{P}$. This fifth drone is also placed in the plane $\mathscr{P}$, at the intersection between the line $\Delta$ and the plane $\mathscr{P}$. We admit that the point $\mathrm{F}(6;-1;3)$ corresponds to this location. Prove that the distance between the starting point D and the plane $\mathscr{P}$ equals $2\sqrt{14}$ hundreds of meters.
The show organizer asks the pilot to send a new drone in the plane (no matter its position in the plane), always starting from point D. Knowing that there are 40 seconds left before the start of the show and that the drone flies in a straight trajectory at $18.6\,\mathrm{m.s}^{-1}$, can the new drone arrive on time?
A municipality decides to replace the traditional July 14 fireworks with a luminous drone show. For drone piloting, space is equipped with an orthonormal reference frame $(\mathrm{O};\vec{\imath},\vec{\jmath},\vec{k})$ whose unit is one hundred meters.
The position of each drone is modeled by a point and each drone is sent from a starting point D with coordinates $(2;5;1)$. It is desired to form figures with drones by positioning them in the same plane $\mathscr{P}$. Three drones are positioned at points $\mathrm{A}(-1;-1;17)$, $\mathrm{B}(4;-2;4)$ and $\mathrm{C}(1;-3;7)$.
\begin{enumerate}
\item Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
In the following, we denote by $\mathscr{P}$ the plane (ABC) and we consider the vector $\vec{n}\begin{pmatrix}2\\-3\\1\end{pmatrix}$.
\item a. Justify that $\vec{n}$ is normal to the plane (ABC).\\
b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $2x - 3y + z - 18 = 0$.
\item The drone pilot decides to send a fourth drone taking as trajectory the line $d$ whose parametric representation is given by
$$d : \left\{\begin{array}{rl} x &= 3t + 2 \\ y &= t + 5 \\ z &= 4t + 1 \end{array},\text{ with } t \in \mathbb{R}.\right.$$
a. Determine a direction vector of the line $d$.\\
b. So that this new drone is also placed in the plane $\mathscr{P}$, determine by calculation the coordinates of point E, the intersection of the line $d$ with the plane $\mathscr{P}$.
\item The drone pilot decides to send a fifth drone along the line $\Delta$ which passes through point $\mathrm{D}$ and which is perpendicular to the plane $\mathscr{P}$. This fifth drone is also placed in the plane $\mathscr{P}$, at the intersection between the line $\Delta$ and the plane $\mathscr{P}$. We admit that the point $\mathrm{F}(6;-1;3)$ corresponds to this location.\\
Prove that the distance between the starting point D and the plane $\mathscr{P}$ equals $2\sqrt{14}$ hundreds of meters.
\item The show organizer asks the pilot to send a new drone in the plane (no matter its position in the plane), always starting from point D. Knowing that there are 40 seconds left before the start of the show and that the drone flies in a straight trajectory at $18.6\,\mathrm{m.s}^{-1}$, can the new drone arrive on time?
\end{enumerate}