Transformations in the Plane

A delivery drone is at position $P(3, -2)$, expressed in miles with respect to its base. The controller sends it an instruction to make a rectilinear displacement defined by the translation vector $\vec{v} = (-5, 4)$. Determine the coordinates of the drone's new position.

A laser cutting machine moves in a straight line over a metal plate from the initial point $A(-1, 5)$ to the final point $B(4, 2)$. Calculate the translation vector that exactly describes this movement.

A crane moves a rectilinear beam horizontally and vertically, whose initial ends are $A(1, 1)$ and $B(5, 3)$. The movement is carried out by a translation defined by the vector $\vec{t} = (0, -4)$. Calculate the coordinates of the new ends $A'$ and $B'$ of the beam.

In the design of a symmetrical garden, a fountain is to be placed at point $F'$ which is the reflection of a statue located at $E(4, 7)$ with respect to a central post located at the origin of coordinates $O(0, 0)$. Find the coordinates where the fountain should be installed.

Two communications satellites, $S_1$ and $S_2$, orbit symmetrically with respect to a central space station. If $S_1$ is at position $(8, -3)$ and the space station is located at $(2, 1)$, calculate the coordinates of the position of satellite $S_2$.

An architect designs a bridge whose arch has a critical anchor point at $A(-3, 5)$. To maintain visual symmetry in a 3D rendering with respect to the water surface, which coincides with the x-axis ($y=0$), they need to locate the virtual reflection of this point underwater. Determine the coordinates of this reflected point.

In a graphic design program, a polygon has been drawn with a key vertex at point $V(5, -2)$. A reflection is applied with respect to the bisector of the first and third quadrants (line $y=x$). Determine analytically the coordinates of the reflected vertex $V'$.

A laser beam is emitted from a projector located at point $P(-2, 4)$. In front of it is a flat mirror placed exactly on the line with equation $r \equiv x - 2y + 5 = 0$. Calculate the coordinates of the virtual point $P'$ (reflection of $P$ in the mirror) from which the reflected light appears to come.

Two forest watchtowers are located at points $T_1(-3, 1)$ and $T_2(5, 5)$. A rectilinear firebreak is to be drawn to act as an axis of symmetry between both towers, so that any point on the firebreak is equidistant from $T_1$ and $T_2$. Calculate the general equation of the line that defines this firebreak.

Two towns, $A(1, 5)$ and $B(7, 2)$, need to connect to a main water pipe that runs in a straight line along the x-axis ($y=0$). A pumping station $P$ is to be built on the main pipe so that the total length of the secondary pipes ($AP + PB$) is minimized. Using the concept of reflection, calculate the exact coordinates of point $P$ and the minimum total length of the pipe.

On a pool table, a cue ball is located at point $B(2, 3)$ and a red ball at $R(8, 6)$. A player wishes to hit the red ball by making the cue ball bounce first off the top cushion of the table, which lies on the line $y = 10$. Assuming that the angle of incidence is equal to the angle of reflection, calculate the coordinates of the exact point on the cushion where the cue ball must hit to achieve their goal.

The path of a conveyor belt follows the line with equation $y = 2x - 1$. Due to space requirements in the factory, the entire belt is translated $3$ units to the left and $2$ units upward. Write the general equation of the new line that defines the translated path.

A cell tower provides coverage in a circular area whose Cartesian equation is $x^2 + y^2 - 4x + 6y - 12 = 0$. Due to interference problems with a nearby building, the company decides to translate the tower (the center of the circle) according to the vector $\vec{v} = (-3, 5)$. Determine the general equation of the new coverage area.

The reflector of a car headlight follows the profile of the parabola $y = x^2 - 6x + 11$. To facilitate the programming of the CNC machining tool, the engineer needs to apply a translation to the parabola so that its vertex coincides exactly with the origin of coordinates. Determine the necessary translation vector and the new equation of the parabola.

The orbit of a comet is projected onto an astronomical plane according to the equation of the ellipse $4x^2 + 9y^2 - 24x + 36y + 36 = 0$. By means of a translation of axes, it is possible to express this equation in its reduced form centered at the origin. Calculate the translation vector that moves the geometric center of the conic to the origin of coordinates and write the equation of the ellipse after this translation.

A railroad track runs following the path of the line $r \equiv 3x + 4y - 12 = 0$. An urban planner wants to lay out a new avenue $s$ that is perfectly symmetrical to the railroad track with respect to a central monument located at $C(1, -1)$. Find the general equation of the avenue $s$.

An agricultural plot is shaped like a triangle with vertices at $A(0, 0)$, $B(4, 0)$ and $C(2, 5)$. The city council acquires the plot and, in exchange, gives the owner another plot obtained by applying a reflection in the vertical line $x = 6$ to the original triangle. Calculate the vertices of the new plot and prove analytically that the area of the plot has not changed.

A ray of light travels along the line $r \equiv 2x - y + 4 = 0$ and hits a flat mirror located on the line $s \equiv x + y - 1 = 0$. Determine the general equation of the line $r'$ containing the reflected ray, knowing that this line is the reflection of $r$ in $s$.

The outline of a circular gear has the equation $x^2 + y^2 - 2x - 4y - 4 = 0$. This gear is reflected in an inclined mirror represented by the line $r \equiv 3x - 4y + 10 = 0$. Determine the general equation of the circle that represents the reflected image of the gear.

A fishing boat makes two consecutive movements on the high seas. First it translates according to the vector $\vec{u} = (2, 6)$ and then according to the vector $\vec{v} = (3, -1)$. If it departed from a port located at coordinates $P(-4, -5)$, determine its final position after both displacements.

In a video game animation, the geometric center of a character located at $A(1, 3)$ undergoes a translation by vector $\vec{t} = (4, -2)$ followed immediately by a reflection across the point $C(2, 0)$. Find the final position of the character and check analytically whether the result would be the same by applying the reflection first and then the translation.

Prove analytically that the composition of two reflections in parallel lines is equivalent to a translation. To do this, consider the lines $r \equiv x - 2y = 0$ and $s \equiv x - 2y + 10 = 0$. Take a generic point $P(x_0, y_0)$, find its reflection $P'$ in $r$, and then the reflection $P''$ of $P'$ in $s$. Determine the translation vector $\vec{v} = \vec{PP''}$ and check that it is perpendicular to the lines and that its modulus is twice the distance between them.

Given the parallel lines $r \equiv 2x + y - 3 = 0$ and $s \equiv 2x + y + 7 = 0$, find the translation vector $\vec{v}$ with the smallest modulus that transforms the line $r$ into the line $s$. Prove analytically that this vector is perpendicular to both lines and that its modulus coincides with the shortest distance between them.

An isometry in the plane transforms point $A(1, 2)$ into $A'(5, 4)$ and point $B(3, -1)$ into $B'(7, 1)$.

1. a) Prove that the distance between $A$ and $B$ is the same as between $A'$ and $B'$.
2. b) Determine whether the described transformation corresponds to a pure translation or a reflection in a point, justifying the answer analytically and finding the corresponding translation vector or center of reflection.