Movements in the plane

A delivery drone is at position $P(3, -2)$, expressed in kilometres 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.

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 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.

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.

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 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.

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.

A mobile phone mast 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 mast (the centre of the circle) according to the vector $\vec{v} = (-3, 5)$. Determine the general equation of the new coverage area.

An agricultural plot is in the shape of a triangle with vertices at $A(0, 0)$, $B(4, 0)$ and $C(2, 5)$. The local council compulsory purchases 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.

In a video game animation, the geometric centre 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.

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 centre of symmetry.