Movements and translations
A drone that moves, a part of a gear that turns, a ray of light that is reflected in a mirror: all three are movements of the plane, and all three can be calculated with very simple formulas in coordinates.
Translations: moving by a vector
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 $A'$?
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 $\vec{PQ}$.
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 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…
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?
Rotations: centre and angle
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…
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…
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 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…
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?
Reflections: in a line or in a point
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…
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 $M'$?
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 $M'$?
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…
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?
Recognising and combining movements
When you know a figure and its image, you can identify which movement was applied by looking at how the coordinates change: if they all add the same amount, it is a translation; if it is $(-x,-y)$, a rotation of 180°; if the sign of just one coordinate changes, a reflection in an axis.
You can also chain movements together: first one, and then the next is applied to the result.
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…
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 $V'$?
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…
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.
Applications: reflections, bounces and unknown centres
Reflections solve problems involving reflection (a ray of light, a billiard ball bouncing off a cushion): the shortest path that touches a "mirror" is found by reflecting one of the points and joining them with a straight line.
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…
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?
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…
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 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…
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 $C$.