For each of the four following propositions, indicate whether it is true or false by justifying the answer. One point is awarded for each correct answer with proper justification. An unjustified answer is not taken into account. An absence of answer is not penalized. Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider the points $\mathrm { A } ( 1 ; 2 ; 5 ) , \mathrm { B } ( - 1 ; 6 ; 4 ) , \mathrm { C } ( 7 ; - 10 ; 8 )$ and $\mathrm { D } ( - 1 ; 3 ; 4 )$.
Proposition 1: The points $\mathrm { A } , \mathrm { B }$ and C define a plane.
We admit that the points $\mathrm { A } , \mathrm { B }$ and D define a plane. Proposition 2: A Cartesian equation of the plane (ABD) is $x - 2 z + 9 = 0$.
Proposition 3: A parametric representation of the line (AC) is $$\left\{ \begin{aligned}
x & = \frac { 3 } { 2 } t - 5 \\
y & = - 3 t + 14 \quad t \in \mathbb { R } \\
z & = - \frac { 3 } { 2 } t + 2
\end{aligned} \right.$$
Let $\mathscr { P }$ be the plane with Cartesian equation $2 x - y + 5 z + 7 = 0$ and $\mathscr { P } ^ { \prime }$ the plane with Cartesian equation $- 3 x - y + z + 5 = 0$. Proposition 4: The planes $\mathscr { P }$ and $\mathscr { P } ^ { \prime }$ are parallel.
\section*{Exercise 3 (4 points)}
For each of the four following propositions, indicate whether it is true or false by justifying the answer.\\
One point is awarded for each correct answer with proper justification. An unjustified answer is not taken into account. An absence of answer is not penalized.\\
Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ).\\
We consider the points $\mathrm { A } ( 1 ; 2 ; 5 ) , \mathrm { B } ( - 1 ; 6 ; 4 ) , \mathrm { C } ( 7 ; - 10 ; 8 )$ and $\mathrm { D } ( - 1 ; 3 ; 4 )$.
\begin{enumerate}
\item Proposition 1: The points $\mathrm { A } , \mathrm { B }$ and C define a plane.
\item We admit that the points $\mathrm { A } , \mathrm { B }$ and D define a plane.\\
Proposition 2: A Cartesian equation of the plane (ABD) is $x - 2 z + 9 = 0$.
\item Proposition 3: A parametric representation of the line (AC) is
$$\left\{ \begin{aligned}
x & = \frac { 3 } { 2 } t - 5 \\
y & = - 3 t + 14 \quad t \in \mathbb { R } \\
z & = - \frac { 3 } { 2 } t + 2
\end{aligned} \right.$$
\item Let $\mathscr { P }$ be the plane with Cartesian equation $2 x - y + 5 z + 7 = 0$ and $\mathscr { P } ^ { \prime }$ the plane with Cartesian equation $- 3 x - y + z + 5 = 0$.\\
Proposition 4: The planes $\mathscr { P }$ and $\mathscr { P } ^ { \prime }$ are parallel.
\end{enumerate}