Exercise 4
In space equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points
$$\mathrm { A } ( - 1 ; - 3 ; 2 ) , \quad \mathrm { B } ( 3 ; - 2 ; 6 ) \quad \text { and } \quad \mathrm { C } ( 1 ; 2 ; - 4 ) .$$
- Prove that the points $\mathrm { A } , \mathrm { B }$ and C define a plane which we will denote $\mathscr { P }$.
- a. Show that the vector $\vec { n } \left( \begin{array} { c } 13 \\ - 16 \\ - 9 \end{array} \right)$ is normal to the plane $\mathscr { P }$. b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $13 x - 16 y - 9 z - 17 = 0$.
We denote $\mathscr { D }$ the line passing through the point $\mathrm { F } ( 15 ; - 16 ; - 8 )$ and perpendicular to the plane $\mathscr { P }$.
3. Give a parametric representation of the line $\mathscr { D }$.
4. We call E the point of intersection of the line $\mathscr { D }$ and the plane $\mathscr { P }$. Prove that the point E has coordinates $( 2 ; 0 ; 1 )$.
5. Determine the exact value of the distance from point F to the plane $\mathscr { P }$. 6. Determine the coordinates of the point(s) on the line $\mathscr { D }$ whose distance to the plane $\mathscr { P }$ is equal to half the distance from point F to the plane $\mathscr { P }$.