Exercises: Relative positions

An observation satellite follows a linear trajectory in space defined by line $r: (x, y, z) = (1, 2, 0) + \lambda(2, -1, 3)$. It is desired to check whether the satellite will pass exactly through a control point located at $P(5, 0, 6)$. Determine whether point $P$ belongs to the satellite's trajectory.

In the design of a smart building, two flat surfaces corresponding to the ceiling and floor of a mechanical room have been defined. The equations of the planes are $\pi_1: 2x - y + 3z - 5 = 0$ and $\pi_2: 4x - 2y + 6z + 8 = 0$. Analyze the relative position of both planes to confirm whether they are parallel.

A laser beam propagates along the line $r: \frac{x-1}{2} = \frac{y+2}{1} = \frac{z}{3}$ and hits a flat sheet of glass whose equation is $\pi: x + y + z - 1 = 0$. Determine the relative position between the laser beam and the sheet. If they intersect, calculate the coordinates of the impact point.

Given the lines $r$ and $s$ expressed by their parametric equations:

$r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 \lambda \end{cases}$ and $s: \begin{cases} x = \mu \\ y = 1 + \mu \\ z = 2 - \mu \end{cases}$

Study the relative position of both by comparing their direction vectors. Are they parallel or do they have different directions?

Check whether the line $r$, defined as the intersection of the planes $x + y = 2$ and $x - y = 0$, is parallel to the coordinate plane $XY$ (the $z = 0$ plane). Justify your answer based on the direction vector of the line.

An architect uses design software to place a structural beam represented by the line $r: \dfrac{x-2}{3} = \dfrac{y+1}{-2} = \dfrac{z-4}{4}$. Determine whether the beam is perpendicular to the facade plane $\pi: 3x - 2y + 4z + 1 = 0$. Justify your answer.

Given the points $A(1, 0, -1)$, $B(2, 1, 0)$ and $C(3, 2, 1)$, determine the relative position of the line passing through $A$ and $B$ with respect to the line passing through $B$ and $C$. Is it the same line (coincident) or are they intersecting lines at the point $B$?

Determine the value of the parameter $k$ so that the planes $\pi_1: x + ky + z - 2 = 0$ and $\pi_2: 2x - 4y + 2z + 1 = 0$ are parallel. For that value of $k$, could the planes be coincident? Justify your answer by observing the constant terms.

Study the relative position of the following lines $r$ and $s$. If they intersect, calculate the coordinates of the point of intersection:

$r: \begin{cases} x = 1 + \lambda \\ y = -2\lambda \\ z = 3 + 2\lambda \end{cases}$

$s: \frac{x-2}{2} = \frac{y+1}{1} = \frac{z-4}{-1}$

Consider plane $\pi: 2x - y + az + 3 = 0$ and line $r$ passing through points $A(1, 0, -1)$ and $B(2, 1, 0)$. Determine the value of parameter $a \in \mathbb{R}$ so that the line is parallel to the plane. For this value of $a$, is the line contained in the plane? Justify your answer.

Two aerial surveillance drones follow straight trajectories defined by lines $r$ and $s$. The control center needs to know if there is any risk of collision (if the trajectories intersect). The equations of the trajectories are:

$r: \frac{x-1}{2} = \frac{y}{1} = \frac{z+2}{3}$

$s: \begin{cases} x + y - z = 0 \\ 2x - y + 3z = 1 \end{cases}$

Determine the relative position of both trajectories.

Discuss, as a function of the real parameter $m$, the relative position of the following three planes:

$\pi_1: mx + y + z = 1$

$\pi_2: x + my + z = m$

$\pi_3: x + y + mz = m^2$

Find the implicit equation of the plane containing line $r: \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z}{3}$ and parallel to line $s: \begin{cases} x = t \\ y = 2t \\ z = 1 + t \end{cases}$. What is the relative position of lines $r$ and $s$?

A pencil of planes has line $r: \begin{cases} 3x - y + z = 2 \\ x + 2y - z = 1 \end{cases}$ as its axis. Find which plane in the pencil passes through point $P(1, 1, 1)$. Once obtained, determine the relative position of this plane with the $OY$ axis.

A laser beam is projected from point $A(2, 3, 5)$ in the direction of vector $\vec{v} = (1, -1, -2)$. The beam hits a metal wall represented by plane $\pi: x + 2y + z = 12$.

1. a) Determine the coordinates of the point of impact.
2. b) Study the relative position of the laser path with line $s: \begin{cases} x = 3 + \mu \\ y = 2 \\ z = 3 - 2\mu \end{cases}$.

Given lines $r$ and $s$, show that they are skew and find the equation of the plane containing $s$ and parallel to $r$:

$r: \frac{x}{1} = \frac{y-1}{2} = \frac{z+1}{1}$

$s: \frac{x-1}{-1} = \frac{y}{3} = \frac{z}{2}$

In an industrial design project, three surfaces are defined by the following planes in space:

$\pi_1: x + ay + z = a$

$\pi_2: ax + y + (a-1)z = 1$

$\pi_3: x + y + z = 1$

Discuss the relative position of the three planes according to the different values of the real parameter $a$. In cases where the three planes intersect in a line (forming a pencil of planes), obtain the parametric equation of this line.

Consider a drone trajectory represented by the line $r$ passing through point $A(1, 2, -1)$ and having direction vector $\vec{v} = (k, 1, 2)$. On the other hand, a solar panel is contained in plane $\pi$ defined by points $P(1, 0, 1)$, $Q(0, 2, 3)$ and $R(2, 1, 0)$.

Determine the value of the parameter $k$ so that the drone's trajectory is parallel to the solar panel. For this value of $k$, is there any line $s$ contained in plane $\pi$ that is perpendicular to $r$? Justify your answer based on the relative position of the elements.

Given the lines $r$ and $s$ defined by:

$r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3\lambda \end{cases}$

$s: \frac{x-4}{2} = \frac{y+1}{m} = z$

1. a) Study the relative position of $r$ and $s$ as a function of the parameter $m$.
2. b) For the value of $m$ that makes the lines intersect, find the equation of the plane $\pi$ containing them.
3. c) If $m=0$, the lines are skew. Find the equation of the line $t$ that passes through the origin and intersects both $r$ and $s$ simultaneously.

An architect designs a structure where three support beams must converge at a point. The faces of the main pillars rest on the planes:

$\pi_1: x + 2y - z = 3$

$\pi_2: 2x - y + z = 1$

$\pi_3: 5x + ay + bz = 5$

Calculate the values of $a$ and $b$ such that:

1. a) The three planes intersect in a single line (the common support beam).
2. b) The three planes have no points in common and intersect pairwise (forming an infinite prism). Explain what relationship the ranks of the coefficient matrix and the augmented matrix must satisfy in this case.

Let line $r$ be defined by the intersection of planes $\sigma_1: x + y = 2$ and $\sigma_2: x - z = 1$. Consider also plane $\pi: 2x + y - z = k$.

1. a) Determine the value of $k$ so that line $r$ is contained in plane $\pi$.
2. b) For a value of $k$ different from the one found in the previous part, line $r$ and plane $\pi$ are parallel. In this situation, calculate the equation of a new line $s$ that is the orthogonal projection of $r$ onto plane $\pi$.
3. c) Study the relative position of this new line $s$ with the $OY$ axis.

Given the lines $r$ and $s$ with equations:

$r: \frac{x-2}{1} = \frac{y+1}{2} = \frac{z}{-1}$

$s: \begin{cases} x - y + 2z = 4 \\ 2x + z = 1 \end{cases}$

1. a) Show analytically that lines $r$ and $s$ are skew.
2. b) Find the equation of the common perpendicular to both lines.
3. c) Determine the relative position of this common perpendicular with the coordinate plane $XZ$.

Consider a family of planes depending on a parameter $\alpha \in \mathbb{R}$:

$\pi_\alpha: \alpha x + y + z = \alpha$

And line $r$ passing through points $M(1, 1, 1)$ and $N(2, 3, 0)$.

1. a) Study the relative position of line $r$ and plane $\pi_\alpha$ according to the values of $\alpha$.
2. b) Is there any value of $\alpha$ for which plane $\pi_\alpha$ contains line $r$?
3. c) For $\alpha = 0$, find the symmetric point of $M$ with respect to plane $\pi_0$ and check whether the line joining $M$ to its symmetric point is intersecting, parallel or coincident with line $r$.

In a stellar coordinate system, two linear radio signals are detected:

$L_1: (x, y, z) = (1, 0, 2) + \lambda(1, -1, 1)$

$L_2: \frac{x}{2} = \frac{y-1}{1} = \frac{z-m}{2}$

1. a) Find the value of $m$ so that the signal paths intersect.
2. b) For the value of $m$ found, calculate the point of impact $I$.
3. c) Consider a satellite located in plane $\pi: x + y + z = 10$. Determine the relative position of line $L_1$ with respect to this plane and calculate, if it exists, the minimum distance between signal $L_1$ and the satellite (represented by the plane).