In space equipped with an orthonormal coordinate system, we consider the points: $$\mathrm{A}(1; 2; 7), \quad \mathrm{B}(2; 0; 2), \quad \mathrm{C}(3; 1; 3), \quad \mathrm{D}(3; -6; 1) \text{ and } \mathrm{E}(4; -8; -4).$$
Show that the points $\mathrm{A}, \mathrm{B}$ and C are not collinear.
Let $\vec{u}(1; b; c)$ be a vector in space, where $b$ and $c$ denote two real numbers. a) Determine the values of $b$ and $c$ such that $\vec{u}$ is a normal vector to the plane (ABC). b) Deduce that a Cartesian equation of the plane (ABC) is: $$x - 2y + z - 4 = 0$$ c) Does the point D belong to the plane (ABC)?
We consider the line $\mathscr{D}$ in space whose parametric representation is: $$\left\{\begin{aligned}
x & = 2t + 3 \\
y & = -4t + 5
\end{aligned}\right.$$
In space equipped with an orthonormal coordinate system, we consider the points:
$$\mathrm{A}(1; 2; 7), \quad \mathrm{B}(2; 0; 2), \quad \mathrm{C}(3; 1; 3), \quad \mathrm{D}(3; -6; 1) \text{ and } \mathrm{E}(4; -8; -4).$$
\begin{enumerate}
\item Show that the points $\mathrm{A}, \mathrm{B}$ and C are not collinear.
\item Let $\vec{u}(1; b; c)$ be a vector in space, where $b$ and $c$ denote two real numbers.\\
a) Determine the values of $b$ and $c$ such that $\vec{u}$ is a normal vector to the plane (ABC).\\
b) Deduce that a Cartesian equation of the plane (ABC) is:
$$x - 2y + z - 4 = 0$$
c) Does the point D belong to the plane (ABC)?
\item We consider the line $\mathscr{D}$ in space whose parametric representation is:
$$\left\{\begin{aligned}
x & = 2t + 3 \\
y & = -4t + 5
\end{aligned}\right.$$
\end{enumerate}