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 center of the garden $C(0, 0)$. Calculate the coordinates of $B'$.

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

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

A rotation of $180^\circ$ is applied to a circular traffic sign with its center at point $C(3, 4)$ and radius $2$. Which exact point on 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 follows 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 their center of symmetry. What are the coordinates of this center?

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 by applying 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 articulates at point $C(2, 1)$. Its gripper is initially located 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 central symmetry with respect to the origin $O(0, 0, 0)$. 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 receiving point $R(1, -1)$. Knowing that the x-coordinate does not change and the y-coordinate of the symmetrical points is equidistant from the mirror, what is the equation of the line that defines the position of the mirror?

In an automated warehouse, a package moves in a straight line from shelf $A(10, 5, 2)$ to the loading zone $B(2, 8, 0)$. Determine the translation vector $\vec{AB}$. If another package located at $C(4, 4, 4)$ makes 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 door located at $P(4, 2)$. To cover a corridor, the camera rotates $90^\circ$ counterclockwise about its axis at $C$. What are the coordinates of the new point $P'$ the camera will point to? (Assume the focal length remains constant).

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

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 ($M''$). What direct translation vector is equivalent to the composition 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, they use the symmetrical point of $B$ with respect to the cushion, let's call it $B'$. Find the coordinates of $B'$ and find the equation of the line joining $A$ and $B'$. At what exact point $P$ on the cushion will the ball rebound?

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

An underwater drone is at position $D(4, -2, -5)$ (the xy-plane represents sea level). First, it ascends vertically a certain distance by applying the translation vector $\vec{v} = (0, 0, 5)$. Next, it emits a sonar pulse that reflects specularly off the seabed, which is completely flat and defined by the equation $z = -10$. What are the coordinates of the symmetrical point of the drone's new position with respect to the seabed?

In the CNC machining of a part, the cutting tool is initially located at point $H(0, 5)$. The program executes two commands: first, the chuck rotates $180^\circ$ about the center $C(2, 3)$, bringing the tool to position $H'$. Immediately after, the carriage translates according to the vector $\vec{u} = (-1, -4)$ until it reaches 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 what type of transformation it is (translation, rotation or reflection) and specify its characteristic elements (vector, center and angle, or axis). Justify your answer analytically.

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

A triangular piece of a historical mosaic with a vertex at $P(2, 6)$ has detached and, upon falling, has rotated exactly $90^\circ$ counterclockwise about an unknown point $C(x, y)$, leaving the vertex in the new position $P'(-4, 4)$. Knowing that in a rotation of $90^\circ$ counterclockwise about 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 center of rotation $C$.