We work in space with an orthonormal coordinate system. We consider the points $\mathrm { A } ( 0 ; 4 ; 1 ) , \mathrm { B } ( 1 ; 3 ; 0 ) , \mathrm { C } ( 2 ; - 1 ; - 2 )$ and $\mathrm { D } ( 7 ; - 1 ; 4 )$.
Prove that the points $\mathrm { A } , \mathrm { B }$ and C are not collinear.
Let $\Delta$ be the line passing through point D with direction vector $\vec { u } ( 2 ; - 1 ; 3 )$. a. Prove that the line $\Delta$ is orthogonal to the plane ( ABC ). b. Deduce a Cartesian equation of the plane (ABC). c. Determine a parametric representation of the line $\Delta$. d. Determine the coordinates of point H, the intersection of the line $\Delta$ and the plane (ABC).
Let $\mathscr { P } _ { 1 }$ be the plane with equation $x + y + z = 0$ and $\mathscr { P } _ { 2 }$ be the plane with equation $x + 4 y + 2 = 0$. a. Prove that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant. b. Verify that the line $d$, the intersection of the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$, has the parametric representation $$\left\{ \begin{array} { l }
x = - 4 t - 2 \\
y = t \\
z = 3 t + 2
\end{array} , t \in \mathbb { R } . \right.$$ c. Are the line $d$ and the plane ( ABC ) secant or parallel?
We work in space with an orthonormal coordinate system.\\
We consider the points $\mathrm { A } ( 0 ; 4 ; 1 ) , \mathrm { B } ( 1 ; 3 ; 0 ) , \mathrm { C } ( 2 ; - 1 ; - 2 )$ and $\mathrm { D } ( 7 ; - 1 ; 4 )$.
\begin{enumerate}
\item Prove that the points $\mathrm { A } , \mathrm { B }$ and C are not collinear.
\item Let $\Delta$ be the line passing through point D with direction vector $\vec { u } ( 2 ; - 1 ; 3 )$.\\
a. Prove that the line $\Delta$ is orthogonal to the plane ( ABC ).\\
b. Deduce a Cartesian equation of the plane (ABC).\\
c. Determine a parametric representation of the line $\Delta$.\\
d. Determine the coordinates of point H, the intersection of the line $\Delta$ and the plane (ABC).
\item Let $\mathscr { P } _ { 1 }$ be the plane with equation $x + y + z = 0$ and $\mathscr { P } _ { 2 }$ be the plane with equation $x + 4 y + 2 = 0$.\\
a. Prove that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant.\\
b. Verify that the line $d$, the intersection of the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$, has the parametric representation
$$\left\{ \begin{array} { l }
x = - 4 t - 2 \\
y = t \\
z = 3 t + 2
\end{array} , t \in \mathbb { R } . \right.$$
c. Are the line $d$ and the plane ( ABC ) secant or parallel?
\end{enumerate}