Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the points $$\mathrm{A}(4; -1; 3), \quad \mathrm{B}(-1; 1; -2), \quad \mathrm{C}(0; 4; 5) \text{ and } \mathrm{D}(-3; -4; 6).$$
a. Verify that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are not collinear. We admit that a Cartesian equation of the plane (ABC) is: $29x + 30y - 17z = 35$. b. Are the points A, B, C, D coplanar? Justify.
Let $P _ { 1 }$ be the perpendicular bisector plane of the segment $[\mathrm{AB}]$. a. Determine the coordinates of the midpoint of the segment $[\mathrm{AB}]$. b. Deduce that a Cartesian equation of $P _ { 1 }$ is: $5x - 2y + 5z = 10$.
We denote by $P _ { 2 }$ the perpendicular bisector plane of the segment $[\mathrm{CD}]$. a. Let M be a point of the plane $P _ { 2 }$ with coordinates $(x; y; z)$. Express $\mathrm{MC}^{2}$ and $\mathrm{MD}^{2}$ as functions of the coordinates of M. Deduce that a Cartesian equation of the plane $P _ { 2 }$ is: $-3x - 8y + z = 10$. b. Justify that the planes $P _ { 1 }$ and $P _ { 2 }$ are secant.
Let $\Delta$ be the line with a parametric representation: $$\left\{ \begin{array} { r l r l }
x & = & -2 - 1.9t \\
y & = & t & \text{ where } t \in \mathbb{R} \\
z & = & 4 + 2.3t
\end{array} \right.$$ Demonstrate that $\Delta$ is the line of intersection of $P _ { 1 }$ and $P _ { 2 }$. We denote by $P _ { 3 }$ the perpendicular bisector plane of the segment $[\mathrm{AC}]$. We admit that a Cartesian equation of the plane $P _ { 3 }$ is: $8x - 10y - 4z = -15$.
Demonstrate that the line $\Delta$ and the plane $P _ { 3 }$ are secant.
Justify that the point of intersection between $\Delta$ and $P _ { 3 }$ is the point H.
Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.\\
We consider the points
$$\mathrm{A}(4; -1; 3), \quad \mathrm{B}(-1; 1; -2), \quad \mathrm{C}(0; 4; 5) \text{ and } \mathrm{D}(-3; -4; 6).$$
\begin{enumerate}
\item a. Verify that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are not collinear.
We admit that a Cartesian equation of the plane (ABC) is: $29x + 30y - 17z = 35$.\\
b. Are the points A, B, C, D coplanar? Justify.
\item Let $P _ { 1 }$ be the perpendicular bisector plane of the segment $[\mathrm{AB}]$.\\
a. Determine the coordinates of the midpoint of the segment $[\mathrm{AB}]$.\\
b. Deduce that a Cartesian equation of $P _ { 1 }$ is: $5x - 2y + 5z = 10$.
\item We denote by $P _ { 2 }$ the perpendicular bisector plane of the segment $[\mathrm{CD}]$.\\
a. Let M be a point of the plane $P _ { 2 }$ with coordinates $(x; y; z)$.
Express $\mathrm{MC}^{2}$ and $\mathrm{MD}^{2}$ as functions of the coordinates of M.\\
Deduce that a Cartesian equation of the plane $P _ { 2 }$ is: $-3x - 8y + z = 10$.\\
b. Justify that the planes $P _ { 1 }$ and $P _ { 2 }$ are secant.
\item Let $\Delta$ be the line with a parametric representation:
$$\left\{ \begin{array} { r l r l }
x & = & -2 - 1.9t \\
y & = & t & \text{ where } t \in \mathbb{R} \\
z & = & 4 + 2.3t
\end{array} \right.$$
Demonstrate that $\Delta$ is the line of intersection of $P _ { 1 }$ and $P _ { 2 }$.
We denote by $P _ { 3 }$ the perpendicular bisector plane of the segment $[\mathrm{AC}]$.\\
We admit that a Cartesian equation of the plane $P _ { 3 }$ is: $8x - 10y - 4z = -15$.
\item Demonstrate that the line $\Delta$ and the plane $P _ { 3 }$ are secant.
\item Justify that the point of intersection between $\Delta$ and $P _ { 3 }$ is the point H.
\end{enumerate}