Exercise 3 (5 points)

We place ourselves in space equipped with an orthonormal coordinate system whose origin is point A. We consider the points $\mathrm{B}(10; -8; 2)$, $\mathrm{C}(-1; -8; 5)$ and $\mathrm{D}(14; 4; 8)$.

1. a. Determine a system of parametric equations for each of the lines (AB) and (CD). b. Verify that the lines (AB) and (CD) are not coplanar.
2. We consider the point I on the line (AB) with abscissa 5 and the point J on the line (CD) with abscissa 4. a. Determine the coordinates of points I and J and deduce the distance IJ. b. Demonstrate that the line (IJ) is perpendicular to the lines (AB) and (CD). The line (IJ) is called the common perpendicular to the lines (AB) and (CD).
3. The purpose of this question is to verify that the distance IJ is the minimum distance between the lines (AB) and (CD). We consider a point $M$ on the line (AB) distinct from point I. We consider a point $M'$ on the line (CD) distinct from point J. a. Justify that the parallel to the line (IJ) passing through point $M'$ intersects the line $\Delta$ (the line parallel to (CD) passing through I) at a point that we will denote $P$. b. Demonstrate that the triangle $MPM'$ is right-angled at $P$. c. Justify that $MM' > IJ$ and conclude.

Exercise 3

Common to all candidates

Two species of turtles endemic to a small island in the Pacific Ocean, green turtles and hawksbill turtles, meet during different breeding episodes on two of the island's beaches to lay eggs. This island, being the convergence point of many turtles, specialists decided to take advantage of this to collect various data on them. They first observed that the corridors used in the ocean by each of the two species to reach the island could be assimilated to rectilinear trajectories. In what follows, space is referred to an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ) with unit 100 meters. The plane ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ) represents the water level and we admit that a point $M ( x ; y ; z )$ with $z < 0$ is located in the ocean. The specialists' model establishes that:

• the trajectory used in the ocean by green turtles is supported by the line $\mathscr { D } _ { 1 }$ whose parametric representation is:

$$\left\{ \begin{aligned} x & = 3 + t \\ y & = 6 t \text { with } t \text { real; } \\ z & = - 3 t \end{aligned} \right.$$

• the trajectory used in the ocean by hawksbill turtles is supported by the line $\mathscr { D } _ { 2 }$ whose parametric representation is:

$$\left\{ \begin{aligned} x & = 10 k \\ y & = 2 + 6 k \text { with } k \text { real; } \\ z & = - 4 k \end{aligned} \right.$$

1. Prove that the two species are never likely to cross before arriving on the island.
2. The objective of this question is to estimate the minimum distance separating these two trajectories. a. Verify that the vector $\vec { n } \left( \begin{array} { c } 3 \\ 13 \\ 27 \end{array} \right)$ is normal to the lines $\mathscr { D } _ { 1 }$ and $\mathscr { D } _ { 2 }$. b. It is admitted that the minimum distance between the lines $\mathscr { D } _ { 1 }$ and $\mathscr { D } _ { 2 }$ is the distance $\mathrm { HH } ^ { \prime }$ where $\overrightarrow { \mathrm { HH } ^ { \prime } }$ is a vector collinear to $\vec { n }$ with H belonging to the line $\mathscr { D } _ { 1 }$ and $\mathrm { H } ^ { \prime }$ belonging to the line $\mathscr { D } _ { 2 }$. Determine an approximate value in meters of this minimum distance. One may use the results below provided by a computer algebra system

\multicolumn{2}{|l|}{$\triangleright$ Computer algebra}
1	\begin{tabular}{ l } Solve $( \{ 10 * k - 3 - t = 3 * l , 2 + 6 * k - 6 * t = 13 * l , - 4 * k + 3 * t = 27 * l \} , \{ k , l , t \} )$
$\rightarrow \left\{ \left\{ k = \frac { 675 } { 1814 } , \ell = \frac { 17 } { 907 } , t = \frac { 603 } { 907 } \right\} \right\}$

\hline \end{tabular}

1. The scientists decide to install a beacon at sea.

It is located at point B with coordinates ( $2 ; 4 ; 0$ ). a. Let $M$ be a point on the line $\mathscr { D } _ { 1 }$.
Determine the coordinates of the point $M$ such that the distance $\mathrm { B} M$ is minimal. b. Deduce the minimum distance, rounded to the nearest meter, between the beacon and the green turtles.

The objective of this exercise is to study the trajectories of two submarines in the diving phase. We consider that these submarines move in a straight line, each at constant speed. At each instant $t$, expressed in minutes, the first submarine is located by the point $S_{1}(t)$ and the second submarine is located by the point $S_{2}(t)$ in an orthonormal reference frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ with unit metre. The plane defined by $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ represents the sea surface. The $z$ coordinate is zero at sea level, negative underwater.

1. We admit that, for every real $t \geqslant 0$, the point $S_{1}(t)$ has coordinates: $$\left\{ \begin{array}{l} x(t) = 140 - 60t \\ y(t) = 105 - 90t \\ z(t) = -170 - 30t \end{array} \right.$$ a. Give the coordinates of the submarine at the beginning of the observation. b. What is the speed of the submarine? c. We place ourselves in the vertical plane containing the trajectory of the first submarine.
   Determine the angle $\alpha$ that the submarine's trajectory makes with the horizontal plane. Give the value of $\alpha$ rounded to 0.1 degree.
2. At the beginning of the observation, the second submarine is located at the point $S_{2}(0)$ with coordinates $(68; 135; -68)$ and reaches after three minutes the point $S_{2}(3)$ with coordinates $(-202; -405; -248)$ at constant speed. At what instant $t$, expressed in minutes, are the two submarines at the same depth?

Two lines $\ell _ { 1 }$ and $\ell _ { 2 }$ in 3-dimensional space are given by $$\ell _ { 1 } = \{ ( t - 9 , - t + 7 , 6 ) \mid t \in \mathbb { R } \} \quad \text{and} \quad \ell _ { 2 } = \{ ( 7 , s + 3 , 3 s + 4 ) \mid s \in \mathbb { R } \}.$$
Questions
(31) The plane passing through the origin and not intersecting either of $\ell _ { 1 }$ and $\ell _ { 2 }$ has equation $ax + by + cz = d$. Write the value of $| a + b + c + d |$ where $a, b, c, d$ are integers with $\gcd = 1$. (32) Let $r$ be the smallest possible radius of a circle that has a point on $\ell _ { 1 }$ as well as a point on $\ell _ { 2 }$. It is given that $r ^ { 2 }$ (i.e., the square of the smallest radius) is an integer. Write the value of $r ^ { 2 }$.

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The shortest distance between $L _ { 1 }$ and $L _ { 2 }$ is
(A) 0
(B) $\frac { 17 } { \sqrt { 3 } }$
(C) $\frac { 41 } { 5 \sqrt { 3 } }$
(D) $\frac { 17 } { 5 \sqrt { 3 } }$

Let $Q$ be the cube with the set of vertices $\left\{ \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) \in \mathbb { R } ^ { 3 } : x _ { 1 } , x _ { 2 } , x _ { 3 } \in \{ 0,1 \} \right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $( 0,0,0 )$ and $( 1,1,1 )$ is in $S$. For lines $\ell _ { 1 }$ and $\ell _ { 2 }$, let $d \left( \ell _ { 1 } , \ell _ { 2 } \right)$ denote the shortest distance between them. Then the maximum value of $d \left( \ell _ { 1 } , \ell _ { 2 } \right)$, as $\ell _ { 1 }$ varies over $F$ and $\ell _ { 2 }$ varies over $S$, is
(A) $\frac { 1 } { \sqrt { 6 } }$
(B) $\frac { 1 } { \sqrt { 8 } }$
(C) $\frac { 1 } { \sqrt { 3 } }$
(D) $\frac { 1 } { \sqrt { 12 } }$

Distance between two parallel planes $2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$ is
(1) $\frac{7}{2}$
(2) $\frac{9}{2}$
(3) $\frac{3}{2}$
(4) $\frac{5}{2}$

A line $l$ passing through origin is perpendicular to the lines $l _ { 1 } : \vec { r } = ( 3 + t ) \hat { \mathrm { i } } + ( - 1 + 2 t ) \hat { \mathrm { j } } + ( 4 + 2 t ) \hat { \mathrm { k } }$ $l _ { 2 } : \vec { r } = ( 3 + 2 s ) \hat { \mathrm { i } } + ( 3 + 2 s ) \hat { \mathrm { j } } + ( 2 + s ) \hat { \mathrm { k } }$ If the co-ordinates of the point in the first octant on $l _ { 2 }$ at a distance of $\sqrt { 17 }$ from the point of intersection of $l$ and $l _ { 1 }$ are $( a , b , c )$, then $18 ( a + b + c )$ is equal to $\underline{\hspace{1cm}}$.

Let the position vectors of two points $P$ and $Q$ be $3 \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$ and $\hat { \mathrm { i } } + 2 \hat { \mathrm { j } } - 4 \widehat { \mathrm { k } }$, respectively. Let $R$ and $S$ be two points such that the direction ratios of lines $P R$ and $Q S$ are $( 4 , - 1,2 )$ and $( - 2,1 , - 2 )$, respectively. Let lines $P R$ and $Q S$ intersect at $T$. If the vector $\overrightarrow { T A }$ is perpendicular to both $\overrightarrow { P R }$ and $\overrightarrow { Q S }$ and the length of vector $\overrightarrow { T A }$ is $\sqrt { 5 }$ units, then the modulus of a position vector of $A$ is :
(1) $\sqrt { 482 }$
(2) $\sqrt { 171 }$
(3) $\sqrt { 5 }$
(4) $\sqrt { 227 }$

If the shortest distance between the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { \lambda }$ and $\frac { x - 2 } { 1 } = \frac { y - 4 } { 4 } = \frac { z - 5 } { 5 }$ is $\frac { 1 } { \sqrt { 3 } }$, then the sum of all possible values of $\lambda$ is:
(1) 16
(2) 6
(3) 12
(4) 15

The shortest distance between the lines $\frac{x+7}{-6} = \frac{y-6}{7} = z$ and $\frac{7-x}{2} = y-2 = z-6$ is
(1) $2\sqrt{29}$
(2) 1
(3) $\sqrt{\frac{37}{2}}$
(4) (truncated)

The shortest distance between the lines $\frac{x-5}{1} = \frac{y-2}{2} = \frac{z-4}{-3}$ and $\frac{x+3}{1} = \frac{y+5}{4} = \frac{z-1}{-5}$ is
(1) $7\sqrt{3}$
(2) $5\sqrt{3}$
(3) $6\sqrt{3}$
(4) $4\sqrt{3}$

Consider the lines $L _ { 1 }$ and $L _ { 2 }$ given by $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z - 2 } { 2 }$ $L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y - 2 } { 2 } = \frac { z - 3 } { 3 }$
A line $L _ { 3 }$ having direction ratios $1 , - 1 , - 2$, intersects $L _ { 1 }$ and $L _ { 2 }$ at the points $P$ and $Q$ respectively. Then the length of line segment $P Q$ is
(1) $2 \sqrt { 6 }$
(2) $3 \sqrt { 2 }$
(3) $4 \sqrt { 3 }$
(4) 4

Let the shortest distance between the lines $L: \frac{x-5}{-2} = \frac{y-\lambda}{0} = \frac{z+\lambda}{1}$, $\lambda \geq 0$ and $L_1: x+1 = y-1 = 4-z$ be $2\sqrt{6}$. If $(\alpha, \beta, \gamma)$ lies on $L$, then which of the following is NOT possible?
(1) $\alpha + 2\gamma = 24$
(2) $2\alpha + \gamma = 7$
(3) $2\alpha - \gamma = 9$
(4) $\alpha - 2\gamma = 19$

The shortest distance between the lines $\frac { x - 4 } { 4 } = \frac { y + 2 } { 5 } = \frac { z + 3 } { 3 }$ and $\frac { x - 1 } { 3 } = \frac { y - 3 } { 4 } = \frac { z - 4 } { 2 }$ is
(1) $6 \sqrt { 3 }$
(2) $2 \sqrt { 6 }$
(3) $6 \sqrt { 2 }$
(4) $3 \sqrt { 6 }$

The shortest distance between the lines $\frac{x+2}{1} = \frac{y}{-2} = \frac{z-5}{2}$ and $\frac{x-4}{1} = \frac{y-1}{2} = \frac{z+3}{0}$ is
(1) 8
(2) 6
(3) 7
(4) 9

If the shortest distance between the lines $\frac { x - \lambda } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { 1 }$ and $\frac { x + 2 } { - 3 } = \frac { y + 5 } { 2 } = \frac { z - 4 } { 4 }$ is $\frac { 44 } { \sqrt { 30 } }$, then the largest possible value of $| \lambda |$ is equal to $\_\_\_\_$

If the square of the shortest distance between the lines $\frac { x - 2 } { 1 } = \frac { y - 1 } { 2 } = \frac { z + 3 } { - 3 }$ and $\frac { x + 1 } { 2 } = \frac { y + 3 } { 4 } = \frac { z + 5 } { - 5 }$ is $\frac { \mathrm { m } } { \mathrm { n } }$, where $\mathrm { m } , \mathrm { n }$ are coprime numbers, then $\mathrm { m } + \mathrm { n }$ is equal to :
(1) 21
(2) 9
(3) 14
(4) 6

Let $\mathrm{L}_1: \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2}$ and $\mathrm{L}_2: \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $\mathrm{L}_1$, then $|5\alpha - 11\beta - 8\gamma|$ equals:
(1) 20
(2) 18
(3) 25
(4) 16

In coordinate space, a plane intersects the plane $x = 0$ and the plane $z = 0$ at lines $L_{1}$ and $L_{2}$, respectively.
Given that $L_{1}$ and $L_{2}$ are parallel, $L_{1}$ passes through the point $(0, 2, -11)$, and $L_{2}$ passes through the point $(8, 21, 0)$,
the distance between $L_{1}$ and $L_{2}$ is $\sqrt{(10-1)(10-2)(10-3)}$. (Express as a simplified radical)