In an orthonormal reference frame ( $\mathrm { O } , \mathrm { I } , \mathrm { J } , \mathrm { K }$ ) with unit 1 cm, we consider the points $\mathrm { A } ( 0 ; - 1 ; 5 )$, $\mathrm { B } ( 2 ; - 1 ; 5 ) , \mathrm { C } ( 11 ; 0 ; 1 ) , \mathrm { D } ( 11 ; 4 ; 4 )$. A point $M$ moves on the line ( AB ) in the direction from A to B at a speed of 1 cm per second. A point $N$ moves on the line (CD) in the direction from C to D at a speed of 1 cm per second. At time $t = 0$ the point $M$ is at A and the point $N$ is at C. We denote $M _ { t }$ and $N _ { t }$ the positions of points $M$ and $N$ after $t$ seconds, $t$ denoting a positive real number. We admit that $M _ { t }$ and $N _ { t }$ have coordinates: $M _ { t } ( t ; - 1 ; 5 )$ and $N _ { t } ( 11 ; 0,8 t ; 1 + 0,6 t )$. Questions 1 and 2 are independent. 1. a. The line $( \mathrm { AB } )$ is parallel to one of the axes $( \mathrm { OI } )$, (OJ) or (OK). Which one? b. The line $( \mathrm { CD } )$ lies in a plane $\mathscr { P }$ parallel to one of the planes $( \mathrm { OIJ } )$, (OIK) or (OJK). Which one? An equation of this plane $\mathscr { P }$ will be given. c. Verify that the line $( \mathrm { AB } )$, orthogonal to the plane $\mathscr { P }$, intersects this plane at the point $\mathrm { E } ( 11 ; - 1 ; 5 )$. d. Are the lines ( AB ) and ( CD ) secant? 2. a. Show that $M _ { t } N _ { t } ^ { 2 } = 2 t ^ { 2 } - 25,2 t + 138$. b. At what time $t$ is the length $M _ { t } N _ { t }$ minimal?
In an orthonormal reference frame ( $\mathrm { O } , \mathrm { I } , \mathrm { J } , \mathrm { K }$ ) with unit 1 cm, we consider the points $\mathrm { A } ( 0 ; - 1 ; 5 )$, $\mathrm { B } ( 2 ; - 1 ; 5 ) , \mathrm { C } ( 11 ; 0 ; 1 ) , \mathrm { D } ( 11 ; 4 ; 4 )$.
A point $M$ moves on the line ( AB ) in the direction from A to B at a speed of 1 cm per second.\\
A point $N$ moves on the line (CD) in the direction from C to D at a speed of 1 cm per second.\\
At time $t = 0$ the point $M$ is at A and the point $N$ is at C.\\
We denote $M _ { t }$ and $N _ { t }$ the positions of points $M$ and $N$ after $t$ seconds, $t$ denoting a positive real number.\\
We admit that $M _ { t }$ and $N _ { t }$ have coordinates: $M _ { t } ( t ; - 1 ; 5 )$ and\\
$N _ { t } ( 11 ; 0,8 t ; 1 + 0,6 t )$.\\
Questions 1 and 2 are independent.
1. a. The line $( \mathrm { AB } )$ is parallel to one of the axes $( \mathrm { OI } )$, (OJ) or (OK). Which one?\\
b. The line $( \mathrm { CD } )$ lies in a plane $\mathscr { P }$ parallel to one of the planes $( \mathrm { OIJ } )$, (OIK) or (OJK). Which one? An equation of this plane $\mathscr { P }$ will be given.\\
c. Verify that the line $( \mathrm { AB } )$, orthogonal to the plane $\mathscr { P }$, intersects this plane at the point $\mathrm { E } ( 11 ; - 1 ; 5 )$.\\
d. Are the lines ( AB ) and ( CD ) secant?
2. a. Show that $M _ { t } N _ { t } ^ { 2 } = 2 t ^ { 2 } - 25,2 t + 138$.\\
b. At what time $t$ is the length $M _ { t } N _ { t }$ minimal?