Exercises: Transformations and translations

A delivery drone is at position $A(3, 5)$ and moves along the vector $\vec{v} = (-4, 2)$. What are the coordinates of its final position $A'$?

On a giant chessboard, a piece moves from the square with coordinates $P(-1, -3)$ to the square $Q(4, 2)$. Determine the components of the translation vector $\vec{PQ}$.

A gardener wants to plant a tree at point $B'$, which must be symmetrical to the tree located at $B(2, -6)$ with respect to the centre of the garden $C(0, 0)$. Calculate the coordinates of $B'$.

In designing a logo, point $M(-4, 7)$ is reflected in the y-axis (axis of ordinates). What are the coordinates of the reflected point $M'$?

A gear component located at point $K(5, 0)$ rotates $90^\circ$ anticlockwise about the origin of coordinates $O(0, 0)$. In what position $K'$ will the component be after the rotation?

A rotation of $180^\circ$ is applied to a circular road sign with centre at point $C(3, 4)$ and radius $2$. Which exact point of the sign remains in the same position (is invariant) after the transformation?

A robot vacuum cleaner moves from its base along the vector $\vec{u} = (2, 5)$ and then along the vector $\vec{w} = (1, -3)$. If the base is at the origin $O(0,0)$, what are the coordinates of its final position, and which single vector $\vec{v}$ represents the total displacement?

Two streetlights are located at points $F_1(-2, 8)$ and $F_2(6, -2)$. An electrical transformer is to be placed at the centre of symmetry of both. What are the coordinates of this centre?

Lucía is programming the movements of a character in a two-dimensional video game. The character is at point $A(2, 5)$ and must be reflected across the horizontal line $y = 1$. Lucía writes in her development notebook: "To reflect across the line $y = 1$, the $x$-coordinate does not change and I subtract the value of the line from the $y$-coordinate, so $y' = 5 - 1 = 4$, leaving the character at point $A'(2, 4)$".

1. a) Without carrying out the correct numerical calculations yet, explain the error in reasoning that Lucía made considering the distance between the original point and the axis of symmetry (on which side of the axis did the supposed image end up?).
2. b) Correct the calculation by applying the analytical expression corresponding to reflections across horizontal lines $y = k$ and determine the correct coordinates of the transformed point $A'$.

An automated guided vehicle at a logistics centre in Zaragoza starts from position $P(5, 2)$. The control software instructs it to perform two consecutive movements: first, a $90^\circ$ counterclockwise rotation about the coordinate origin $(0, 0)$; second, a translation along the vector $\vec{v} = (-3, 4)$.

1. a) Estimate: Before performing the algebraic operations, deduce with reasoning in which quadrant of the plane the vehicle will end up immediately after the rotation. Next, predict what sign the abscissa and ordinate of the final position will have after applying the translation $\vec{v}$.
2. b) Check: Calculate step by step the coordinates of the intermediate point after the rotation and those of the final point after the translation. Check whether the numerical result confirms the predictions you made in the estimation.

In architectural software (CAD), the plan of a window is defined by the polygon with vertices $A(1, 1)$, $B(4, 1)$, $C(4, 3)$ and $D(1, 3)$. The designer decides to translate the window using the vector $\vec{v} = (-3, 2)$. Calculate the coordinates of the new vertices $A'$, $B'$, $C'$ and $D'$.

In a textile factory, a machine folds a fabric along the horizontal line $y = 2$. If a decorative pattern is printed at point $P(3, -1)$, what will be the coordinates of the pattern $P'$ once the fabric is folded?

A robotic arm pivots at point $C(2, 1)$. Its gripper is initially at $P(5, 3)$. If the arm performs a rotation of $180^\circ$ about the joint $C$, what will be the new coordinates $P'$ of the gripper?

A 3D printing program first translates a vertex of a solid using the vector $\vec{v} = (0, 0, 4)$ and then applies a point reflection about the origin of coordinates $O(0, 0, 0)$. If the original vertex was at position $V(2, -3, 1)$, what is its final position $V'$?

A laser beam is emitted from point $E(1, 5)$ and, after reflecting off a flat mirror parallel to the x-axis, hits the receiver point $R(1, -1)$. Knowing that the x-coordinate does not change and the y-coordinate of the reflected points is equidistant from the mirror, what is the equation of the line defining the position of the mirror?

In an automated warehouse, a parcel moves in a straight line from shelf $A(10, 5, 2)$ to loading area $B(2, 8, 0)$. Determine the translation vector $\vec{AB}$. If another parcel located at $C(4, 4, 4)$ performs exactly the same movement, what will be the coordinates of its final position $C'$?

A security camera mounted on a pole at $C(1, 2)$ initially focuses on a doorway located at $P(4, 2)$. To cover a corridor, the camera rotates $90^\circ$ anticlockwise about its axis at $C$. What are the coordinates of the new point $P'$ at which the camera will point? (Assume the focal length remains constant).

A sailboat moves across the sea driven by the wind, giving it a displacement vector of $\vec{v} = (12, -5)$ km. Simultaneously, a sea current drags it according to vector $\vec{c} = (-3, -2)$ km. If the sailboat departed from the port at $P(-4, 10)$, what is its final position $P'$ and what is its straight-line distance from the starting point?

The State Meteorological Agency (AEMET) has a radar station located at point $C(3, 1)$ (coordinates in tens of kilometres). The radar detects the core of a storm at position $P(7, 4)$. Due to the wind pattern, the air mass rotates $90^\circ$ counterclockwise around station $C$ and, immediately afterwards, moves along the translation vector $\vec{u} = (-2, 3)$.

1. a) Describe the three geometric steps required to perform a rotation when the centre does not coincide with the coordinate origin.
2. b) Find the exact coordinates of the final position of the storm core after both movements are completed.
3. c) Calculate the straight-line distance, in real kilometres, from the storm to radar $C$ at its final position (remember that $1\text{ unit} = 10\text{ km}$).

For the redesign of a sports club logo, two independent geometric transformations are applied to the coordinate plane:

• An axial symmetry with respect to an unknown vertical line $r: x = k$, which transforms vertex $A(-3, 4)$ into point $A'(7, 4)$.

• A translation along an unknown vector $\vec{w}$, which moves mark $B(2, -1)$ to position $B'(-1, 5)$.

1. a) Determine the value of parameter $k$ defining the axis of symmetry and find the components of the translation vector $\vec{w}$.
2. b) Solve the inverse problem: find out what the coordinates of the original point $P(x, y)$ were, knowing that after applying first the translation $\vec{w}$ and then the symmetry with respect to the line $x = k$, its resulting final position is point $(1, 2)$.

A ray of light passes through a thick crystal and undergoes a double internal reflection. First, it undergoes a reflection across the vertical line $x = 1$, and then another reflection across the line $x = 4$. If an impurity in the crystal is at point $M(-2, 3)$, find its position after the first reflection ($M'$) and after the second ($M''$). Which direct translation vector is equivalent to the composite of these two reflections?

On a billiard table, a ball is at position $A(2, 3)$. A player wants to hit it so that it strikes the right cushion (located along the line $x = 8$) and, after rebounding, hits another ball located at $B(4, 7)$. To calculate the trajectory, use the reflection of point $B$ across the cushion, let us call it $B'$. Find the coordinates of $B'$ and find the equation of the line connecting $A$ with $B'$. At what exact point $P$ on the cushion will the ball rebound?

A square has its geometric centre at the origin $O(0, 0)$ and one of its vertices is $V_1(3, 3)$. To find the remaining vertices, successive clockwise rotations of $90^\circ$ about the origin are applied, obtaining $V_2$, $V_3$ and $V_4$. Calculate the coordinates of the four vertices and verify analytically that the distance between $V_1$ and $V_2$ is exactly equal to the distance between $V_2$ and $V_3$.

In CNC machining of a part, the cutting tool is initially at point $H(0, 5)$. The program executes two commands: first, the chuck rotates $180^\circ$ about the centre $C(2, 3)$, moving the tool to position $H'$. Immediately afterwards, the carriage translates along the vector $\vec{u} = (-1, -4)$ until reaching the final position $H''$. Determine the exact coordinates of $H'$ and $H''$.

In a study of geometric transformations, the vertices of a triangle are $A(1, 2)$, $B(3, 1)$ and $C(2, 4)$. After applying a single transformation in the plane, the new vertices are $A'(-1, -2)$, $B'(-3, -1)$ and $C'(-2, -4)$. Identify the type of transformation (translation, rotation or reflection) and specify its characteristic elements (vector, centre and angle, or axis of reflection). Justify your answer analytically.

A straight water supply pipe passes through the points $A(0, 4)$ and $B(4, 0)$. For design redundancy reasons, a second pipe is constructed that is symmetrical to the first with respect to the x-axis (axis of abscissae). Find the coordinates of the reflected points $A'$ and $B'$, and determine the explicit equation of the line defining the path of this second pipe.

A triangular piece of a historic mosaic with a vertex at $P(2, 6)$ has come loose and, on falling, has rotated exactly $90^\circ$ anticlockwise around an unknown point $C(x, y)$, leaving the vertex in the new position $P'(-4, 4)$. Given that in an anticlockwise rotation of $90^\circ$ around the origin a vector $(a, b)$ is transformed into $(-b, a)$, set up the equations corresponding to the transformation of the vector $\vec{CP}$ into $\vec{CP'}$ and solve the system to find the coordinates of the centre of rotation $C$.

In a graphics library for programming industrial robot trajectories, it is required to transform any point $P(x, y)$ via central symmetry with respect to a pivot point $C(a, b)$. The technical team considers two different algorithms:

• Method 1: Apply directly the analytical formula for central symmetry with respect to a point: $P' = 2C - P$.

• Method 2: Compose two consecutive perpendicular axial symmetries: first reflect $P(x, y)$ with respect to the vertical line $x = a$ to obtain an intermediate point $P_1$, and then reflect $P_1$ with respect to the horizontal line $y = b$ to obtain $P'$.

1. a) Algebraically derive the final coordinates $(x', y')$ obtained by both methods for a generic point $P(x, y)$ with centre $C(a, b)$. Rigorously prove whether both methods are mathematically equivalent.
2. b) Justify whether the order in which the two symmetries are applied in Method 2 affects the final result.
3. c) If you must implement this calculation on a microcontroller aimed at minimising the total number of real-time arithmetic operations, compare the computational cost of both methods and justify which one you would choose.

A Salvamento Marítimo helicopter based at coastal point $A(2, 6)$ must carry out an emergency intervention. The mission requires it to first descend to the shoreline level (represented by the horizontal line $y = 1$) to collect rescue equipment at a point $P(x_0, 1)$, and immediately proceed to a vessel in distress located at $B(8, 4)$ (coordinates are expressed in nautical miles). To minimise fuel consumption, the trajectory must be optimised so that the total distance travelled $d = AP + PB$ is as short as possible.

1. a) Find the coordinates of the symmetric point of $B$ with respect to the shoreline $y = 1$, which we will call $B'$.
2. b) Prove with reasoning, relying on the properties of axial symmetry and the triangle inequality, why the distance $AP + PB$ is equal to the length of the path $AP + PB'$, and deduce why the optimal point $P$ must necessarily belong to the straight line segment joining $A$ and $B'$.
3. c) Obtain the equation of the line passing through points $A$ and $B'$, and use it to determine the exact coordinates of the contact point $P(x_0, 1)$ on the shoreline.

Topic Challenge. How does the wind influence the design of corridors for emergency drones?

1. Real data: Check the AEMET website for the wind speed and direction for 6 consecutive hours at your nearest weather station. Model each record as a vector in the plane $\vec{v}_i = (x_i, y_i)$.

2. Statistical analysis: In a spreadsheet, determine the measures of central tendency (mean and median) and dispersion (variance and range) of the horizontal and vertical components to summarize the dispersion of the airflow.

3. Construction in GeoGebra: Fix your school at point $A(x_0, y_0)$. Apply a translation to $A$ using the mean vector $\vec{u} = (\bar{x}, \bar{y})$ to obtain $A'$. Find the equation of the line $r$ connecting both points, a line $s$ perpendicular to $r$ (crosswind runway), and the lines parallel to the Cartesian axes that pass through $A$.

4. Final product: Submit a justified technical proposal with a screenshot of the geometric model, arguing for the safest orientation for takeoff.

Assessment criteria:

• Precise calculation of vectors and rigor in calculating measures of central tendency and dispersion.

• Algebraic and geometric correctness in vector operations, translations, and parallel/perpendicular lines.

• Report quality and technical justification based on the simulation.