Exercises: Transformations in the plane

A delivery drone is at position $P(3, -2)$, expressed in kilometers relative to its base. The controller sends it an instruction to make a straight-line 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 on a metal sheet from the initial point $A(-1, 5)$ to the final point $B(4, 2)$. Calculate the translation vector that describes this movement exactly.

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 $O(0, 0)$. Find the coordinates where the fountain should be installed.

A fishing boat undergoes two consecutive movements on the open sea. 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.

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 at $(2, 1)$, calculate the coordinates of the position of satellite $S_2$.

An architect designs a bridge whose arch has a critical anchoring 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 under the water. Determine the coordinates of this symmetrical point.

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

The trajectory 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 upwards. Write the general equation of the new line that defines the translated trajectory.

In a graphic design program, a polygon with a key vertex at point $V(5, -2)$ has been drawn. An axial symmetry (reflection) across the angle bisector of the first and third quadrants (line $y=x$) is applied. Analytically determine the coordinates of the reflected vertex $V'$.

A cell phone tower provides coverage in a circular area whose Cartesian equation is $x^2 + y^2 - 4x + 6y - 12 = 0$. Due to interference issues 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.

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

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 point symmetry (point reflection) with respect to point $C(2, 0)$. Find the final position of the character and verify analytically whether the result would be the same by applying the point symmetry first and then the translation.

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

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 translate the parabola so that its vertex coincides exactly with the origin. Determine the required translation vector and the new equation of the parabola.

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

An agricultural plot is in the shape of a triangle with vertices at $A(0, 0)$, $B(4, 0)$ and $C(2, 5)$. The municipality expropriates the plot and, in exchange, gives the owner another plot obtained by applying a reflection across 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 plane 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 $r$ across $s$.

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

Prove analytically that the composition of two reflections across 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 $P'$ across $r$, and then the reflection $P''$ of $P'$ across $s$. Determine the translation vector $\vec{v} = \vec{PP''}$ and check that it is perpendicular to the lines and that its magnitude is twice the distance between them.

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 translating the axes, it is possible to express this equation in its canonical form centered at the origin. Calculate the translation vector that moves the geometric center of the conic to the origin, and write the equation of the ellipse after this translation.

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 representing the reflected image of the gear.

On a billiard 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 having the cue ball bounce first off the top cushion of the table, which lies along the line $y = 10$. Assuming 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 strike to achieve its objective.

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

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

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