Exercise 2 — 5 points Theme: geometry in space Space is equipped with an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• $d_1$ the line passing through point $H(2; 3; 0)$ with direction vector $\vec{u}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$;
• $d_2$ the line with parametric representation:

$$\left\{\begin{aligned} x &= 2k - 3\\ y &= k\\ z &= 5 \end{aligned}\quad\text{where }k\text{ describes }\mathbb{R}.\right.$$ The purpose of this exercise is to determine a parametric representation of a line $\Delta$ that is perpendicular to both lines $d_1$ and $d_2$.

1. a. Determine a direction vector $\vec{v}$ of line $d_2$. b. Prove that lines $d_1$ and $d_2$ are not parallel. c. Prove that lines $d_1$ and $d_2$ are not intersecting. d. What is the relative position of lines $d_1$ and $d_2$?
2. a. Verify that the vector $\vec{w}\left(\begin{array}{c}-1\\2\\3\end{array}\right)$ is orthogonal to both $\vec{u}$ and $\vec{v}$. b. We consider the plane $P$ passing through point $H$ and directed by vectors $\vec{u}$ and $\vec{w}$. We admit that a Cartesian equation of this plane is: $$5x + 4y - z - 22 = 0.$$ Prove that the intersection of plane $P$ and line $d_2$ is the point $M(3; 3; 5)$.
3. Let $\Delta$ be the line with direction vector $\vec{w}$ passing through point $M$.
   A parametric representation of $\Delta$ is therefore given by: $$\left\{\begin{array}{l} x = -r + 3\\ y = 2r + 3\\ z = 3r + 5 \end{array}\text{ where }r\text{ describes }\mathbb{R}.\right.$$ a. Justify that lines $\Delta$ and $d_1$ are perpendicular at a point $L$ whose coordinates you will determine. b. Explain why line $\Delta$ is a solution to the problem posed.

Exercise 4

The objective of this exercise is to determine the distance between two non-coplanar lines. By definition, the distance between two non-coplanar lines in space, $( d _ { 1 } )$ and $( d _ { 2 } )$ is the length of the segment $[\mathrm { EF }]$, where E and F are points belonging respectively to $\left( d _ { 1 } \right)$ and to $( d _ { 2 } )$ such that the line (EF) is orthogonal to $( d _ { 1 } )$ and $( d _ { 2 } )$. The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. Let $\left( d _ { 1 } \right)$ be the line passing through $\mathrm { A } ( 1 ; 2 ; - 1 )$ with direction vector $\overrightarrow { u _ { 1 } } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$ and $\left( d _ { 2 } \right)$ the line with parametric representation: $\left\{ \begin{array} { l } x = 0 \\ y = 1 + t \\ z = 2 + t \end{array} , t \in \mathbb { R } \right.$.

1. Give a parametric representation of the line $\left( d _ { 1 } \right)$.
2. Prove that the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$ are non-coplanar.
3. Let $\mathscr { P }$ be the plane passing through A and directed by the non-collinear vectors $\overrightarrow { u _ { 1 } }$ and $\vec { w } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$. Justify that a Cartesian equation of the plane $\mathscr { P }$ is: $- 2 x + y + 5 z + 5 = 0$.
4. a. Without seeking to calculate the coordinates of the intersection point, justify that the line $( d _ { 2 } )$ and the plane $\mathscr { P }$ are secant. b. We denote F the intersection point of the line $( d _ { 2 } )$ and the plane $\mathscr { P }$. Verify that the point F has coordinates $\left( 0 ; - \frac { 5 } { 3 } ; - \frac { 2 } { 3 } \right)$. Let $( \delta )$ be the line passing through F with direction vector $\vec { w }$. It is admitted that the lines $( \delta )$ and $( d _ { 1 } )$ are secant at a point E with coordinates $\left( - \frac { 2 } { 3 } ; - \frac { 4 } { 3 } ; - 1 \right)$.
5. a. Justify that the distance EF is the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$. b. Calculate the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$.

In coordinate space, there are two mutually perpendicular planes $\alpha$ and $\beta$. For two points $\mathrm { A }$ and $\mathrm { B }$ on plane $\alpha$, $\overline { \mathrm { AB } } = 3 \sqrt { 5 }$, and line AB is parallel to plane $\beta$. The distance between point A and plane $\beta$ is 2, and the distance between a point P on plane $\beta$ and plane $\alpha$ is 4. Find the area of triangle PAB. [4 points]

In coordinate space, there are three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ not on the same line. For a plane $\alpha$ satisfying the following conditions, let $d ( \alpha )$ be the minimum distance among the distances from each point $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to plane $\alpha$.
(a) Plane $\alpha$ intersects segment AC and also intersects segment BC.
(b) Plane $\alpha$ does not intersect segment AB.
Among planes $\alpha$ satisfying the above conditions, let $\beta$ be the plane where $d ( \alpha )$ is maximized. Which of the following statements in are correct? [4 points]
$\text{ㄱ}$. Plane $\beta$ is perpendicular to the plane passing through the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$. $\text{ㄴ}$. Plane $\beta$ passes through the midpoint of segment AC or the midpoint of segment BC. $\text{ㄷ}$. When the three points are $\mathrm { A } ( 2,3,0 ) , \mathrm { B } ( 0,1,0 ) , \mathrm { C } ( 2 , - 1,0 )$, $d ( \beta )$ equals the distance between point B and plane $\beta$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the coordinate plane, for a parallelogram OACB with $\overline { \mathrm { OA } } = \sqrt { 2 } , \overline { \mathrm { OB } } = 2 \sqrt { 2 }$ and $\cos ( \angle \mathrm { AOB } ) = \frac { 1 } { 4 }$, point P satisfies the following conditions. (가) $\overrightarrow { \mathrm { OP } } = s \overrightarrow { \mathrm { OA } } + t \overrightarrow { \mathrm { OB } } ( 0 \leq s \leq 1, 0 \leq t \leq 1 )$ (나) $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OB } } + \overrightarrow { \mathrm { BP } } \cdot \overrightarrow { \mathrm { BC } } = 2$
For a point X moving on a circle centered at O and passing through point A, let $M$ and $m$ be the maximum and minimum values of $| 3 \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OX } } |$ respectively. When $M \times m = a \sqrt { 6 } + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Given that $a$ and $b$ are rational numbers.) [4 points]

As shown in the figure, in the triangular pyramid $P - A B C$, $A B = B C = 2 \sqrt { 2 }$, $P A = P B = P C = A C = 4$, and $O$ is the midpoint of $A C$.
(1) Prove: $P O \perp$ plane $A B C$;
(2) If point $M$ is on edge $B C$ such that $M C = 2 M B$, find the distance from point $C$ to plane $P O M$.

A fountain mounted on a pole consists of a marble sphere resting in a bronze bowl. The marble sphere touches the four inner walls of the bronze bowl at exactly one point each. The bronze bowl is described in the model by the lateral faces of the pyramid ABCDS, the marble sphere by a sphere with center $M ( 0 | 0 | 4 )$ and radius $r$. The $x _ { 1 } x _ { 2 }$-plane of the coordinate system represents the horizontally running ground in the model; one unit of length corresponds to one decimeter in reality.
Determine the diameter of the marble sphere to the nearest centimeter.

135- What is the length of the common perpendicular of the two lines $$\frac{x-1}{1} = \frac{y+2}{-1} = \frac{z}{3} \quad \text{and} \quad \begin{cases} x = 2y - 1 \\ z = 3y - 2 \end{cases}$$?
(1) $\sqrt{3}$ (2) $\sqrt{6}$ (3) $2\sqrt{3}$ (4) $2\sqrt{6}$

135- The shortest distance between the two lines $\dfrac{x-1}{3} = -y + 4 = \dfrac{z}{5}$ and $\begin{cases} x = 2 \\ y = 5 \end{cases}$ is which of the following?
(1) $\dfrac{3}{\sqrt{10}}$ (2) $\dfrac{4}{\sqrt{10}}$ (3) $\sqrt{10}$ (4) $2\sqrt{5}$

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 distance of the point $( 1,1,1 )$ from the plane passing through the point $( - 1 , - 2 , - 1 )$ and whose normal is perpendicular to both the lines $L _ { 1 }$ and $L _ { 2 }$ is
(A) $\frac { 2 } { \sqrt { 75 } }$
(B) $\frac { 7 } { \sqrt { 75 } }$
(C) $\frac { 13 } { \sqrt { 75 } }$
(D) $\frac { 23 } { \sqrt { 75 } }$

If the distance between the plane $\mathrm { Ax } - 2 \mathrm { y } + \mathrm { z } = \mathrm { d }$ and the plane containing the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 4 }$ and $\frac { x - 2 } { 3 } = \frac { y - 3 } { 4 } = \frac { z - 4 } { 5 }$ is $\sqrt { 6 }$, then $| d |$ is

Match the statements in Column-I with the values in Column-II.
Column I
A) A line from the origin meets the lines $\frac { x - 2 } { 1 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 1 }$ and $\frac { x - \frac { 8 } { 3 } } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 1 }$ at $P$ and $Q$ respectively. If length $\mathrm { PQ } = d$, then $d ^ { 2 }$ is
B) The values of $x$ satisfying $\tan ^ { - 1 } ( x + 3 ) - \tan ^ { - 1 } ( x - 3 ) = \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ are
C) Non-zero vectors $\vec { a } , \vec { b }$ and $\vec { c }$ satisfy $\vec { a } \cdot \vec { b } = 0$, $( \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } ) \cdot ( \overrightarrow { \mathrm { b } } + \overrightarrow { \mathrm { c } } ) = 0$ and $2 | \overrightarrow { \mathrm {~b} } + \overrightarrow { \mathrm { c } } | = | \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } |$. If $\vec { a } = \mu \vec { b } + 4 \vec { c }$, then the possible values of $\mu$ are
D) Let f be the function on $[ - \pi , \pi ]$ given by $f ( 0 ) = 9$ and $f ( x ) = \sin \left( \frac { 9 x } { 2 } \right) / \sin \left( \frac { x } { 2 } \right)$ for $x \neq 0$. The value of $\frac { 2 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) d x$ is
Column II p) $-4$ q) $0$ r) $4$ s) $-1$ (or as given in paper) t) $6$

A line $l$ passing through the origin is perpendicular to the lines $l _ { 1 } : ( 3 + t ) \hat { i } + ( - 1 + 2 t ) \hat { j } + ( 4 + 2 t ) \hat { k } , - \infty < t < \infty$ $l _ { 2 } : ( 3 + 2 s ) \hat { i } + ( 3 + 2 s ) \hat { j } + ( 2 + s ) \hat { k } , - \infty < s < \infty$ Then, the coordinate(s) of the point(s) on $l _ { 2 }$ at a distance of $\sqrt { 17 }$ from the point of intersection of $l$ and $l _ { 1 }$ is (are)
(A) $\left( \frac { 7 } { 3 } , \frac { 7 } { 3 } , \frac { 5 } { 3 } \right)$
(B) $( - 1 , - 1,0 )$
(C) $( 1,1,1 )$
(D) $\left( \frac { 7 } { 9 } , \frac { 7 } { 9 } , \frac { 8 } { 9 } \right)$

In $\mathbb { R } ^ { 3 }$, consider the planes $P _ { 1 } : y = 0$ and $P _ { 2 } : x + z = 1$. Let $P _ { 3 }$ be a plane, different from $P _ { 1 }$ and $P _ { 2 }$, which passes through the intersection of $P _ { 1 }$ and $P _ { 2 }$. If the distance of the point $( 0,1,0 )$ from $P _ { 3 }$ is 1 and the distance of a point $( \alpha , \beta , \gamma )$ from $P _ { 3 }$ is 2, then which of the following relations is (are) true?
(A) $2 \alpha + \beta + 2 \gamma + 2 = 0$
(B) $2 \alpha - \beta + 2 \gamma + 4 = 0$
(C) $2 \alpha + \beta - 2 \gamma - 10 = 0$
(D) $2 \alpha - \beta + 2 \gamma - 8 = 0$

Let $\ell _ { 1 }$ and $\ell _ { 2 }$ be the lines $\vec { r } _ { 1 } = \lambda ( \hat { i } + \hat { j } + \hat { k } )$ and $\vec { r } _ { 2 } = ( \hat { j } - \hat { k } ) + \mu ( \hat { i } + \hat { k } )$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell _ { 1 }$. For a plane $H$, let $d ( H )$ denote the smallest possible distance between the points of $\ell _ { 2 }$ and $H$. Let $H _ { 0 }$ be a plane in $X$ for which $d \left( H _ { 0 } \right)$ is the maximum value of $d ( H )$ as $H$ varies over all planes in $X$.
Match each entry in List-I to the correct entries in List-II.
List-I
(P) The value of $d \left( H _ { 0 } \right)$ is
(Q) The distance of the point $( 0,1,2 )$ from $H _ { 0 }$ is
(R) The distance of origin from $H _ { 0 }$ is
(S) The distance of origin from the point of intersection of planes $y = z , x = 1$ and $H _ { 0 }$ is
List-II
(1) $\sqrt { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) 0
(4) $\sqrt { 2 }$
(5) $\frac { 1 } { \sqrt { 2 } }$
The correct option is:
(A) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 5 )$ $( S ) \rightarrow ( 1 )$
(B) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 1 )$
(C) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 2 )$
(D) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 2 )$

The distance of the point $(1, -5, 9)$ from the plane $x - y + z = 5$ measured along the line $x = y = z$ is:
(1) $3\sqrt{10}$
(2) $10\sqrt{3}$
(3) $\frac{10}{\sqrt{3}}$
(4) $\frac{20}{3}$

The distance of the point $(1, -5, 9)$ from the plane $x - y + z = 5$ measured along the line $x = y = z$ is: (1) $3\sqrt{10}$ (2) $10\sqrt{3}$ (3) $\frac{10}{\sqrt{3}}$ (4) $\frac{20}{3}$

The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$, having normal perpendicular to both the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 2 } = \frac { z - 4 } { 3 } \quad \text{and} \quad \frac { x - 2 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z + 7 } { - 1 }$$ is:
(1) $\frac { 10 } { \sqrt { 74 } }$
(2) $\frac { 20 } { \sqrt { 74 } }$
(3) $\frac { 5 } { \sqrt { 83 } }$
(4) $\frac { 10 } { \sqrt { 83 } }$

If $L _ { 1 }$ is the line of intersection of the planes $2 x - 2 y + 3 z - 2 = 0 , x - y + z + 1 = 0$ and $L _ { 2 }$ is the line of intersection of the planes $x + 2 y - z - 3 = 0,3 x - y + 2 z - 1 = 0$, then the distance of the origin from the plane, containing the lines $L _ { 1 }$ and $L _ { 2 }$ is
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 4 \sqrt { 2 } }$
(3) $\frac { 1 } { 3 \sqrt { 2 } }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$

The length of the perpendicular from the point $(2,-1,4)$ on the straight line $\frac{x+3}{10} = \frac{y-2}{-7} = \frac{z}{1}$ is
(1) greater than 3 but less than 4
(2) greater than 4
(3) less than 2
(4) greater than 2 but less than 3

Let $P$ be the plane, which contains the line of intersection of the planes, $x + y + z - 6 = 0$ and $2 x + 3 y + z + 5 = 0$ and it is perpendicular to the $x y$-plane. Then the distance of the point $( 0,0,256 )$ from $P$ is equal to:
(1) $205 \sqrt { 5 }$ units
(2) $\frac { 17 } { \sqrt { 5 } }$ units
(3) $\frac { 11 } { \sqrt { 5 } }$ units
(4) $63 \sqrt { 5 }$ units

The distance of the point $( 1 , - 2,3 )$ from the plane $x - y + z = 5$ measured parallel to the line $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { - 6 }$ is:
(1) $\frac { 7 } { 5 }$
(2) 1
(3) $\frac { 1 } { 7 }$
(4) 7

Let the foot of the perpendicular from the point $( 1,2,4 )$ on the line $\frac { x + 2 } { 4 } = \frac { y - 1 } { 2 } = \frac { z + 1 } { 3 }$ be $P$. Then the distance of $P$ from the plane $3 x + 4 y + 12 z + 23 = 0$ is
(1) $\frac { 50 } { 13 }$
(2) $\frac { 63 } { 13 }$
(3) $\frac { 65 } { 13 }$
(4) 4

The shortest distance between the lines $\frac { x - 3 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 1 } { - 1 }$ and $\frac { x + 3 } { 2 } = \frac { y - 6 } { 1 } = \frac { z - 5 } { 3 }$ is
(1) $\frac { 18 } { \sqrt { 5 } }$
(2) $\frac { 22 } { 3 \sqrt { 5 } }$
(3) $\frac { 46 } { 3 \sqrt { 5 } }$
(4) $6 \sqrt { 3 }$

Let $P$ be the plane containing the straight line $\frac { x - 3 } { 9 } = \frac { y + 4 } { - 1 } = \frac { z - 7 } { - 5 }$ and perpendicular to the plane containing the straight lines $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 5 }$ and $\frac { x } { 3 } = \frac { y } { 7 } = \frac { z } { 8 }$. If $d$ is the distance of $P$ from the point $(2,-5,11)$, then $d ^ { 2 }$ is equal to
(1) $\frac { 147 } { 2 }$
(2) 96
(3) $\frac { 32 } { 3 }$
(4) 54