In maths, the meaning of rotation is the circular motion of a thing around a centre or axis. One of the best real-life examples of rotation is 'earth's rotation on its axis.' Here, we will discuss rotation in detail, including its definition, formula, matrix, symmetry, and examples.

Rotation in mathematics is a transformation that rotates a figure about a fixed point known as the centre of rotation. In rotation, there is a central point that remains fixed, and everything else rotates around that point. In other words, rotation can be described as the motion of a rigid body about a fixed point. The object and its rotation are the same in size and shape. Hence, the figures can be turned in different directions. In general, a rotation can be clockwise or counterclockwise. Note that a clockwise rotation has a negative magnitude, whereas a counterclockwise has a positive magnitude.

In other words, a rotation is a turn of a shape and is described by the direction of the turn, the angle of rotation, and the centre of rotation, a point around which a shape rotates. Each point in the shape must stay at an equal distance from the centre of rotation.

Rotations in the Coordinate Plane

Note that the rotations on the coordinate planes are considered as counterclockwise unless otherwise stated. Let's see some cases:

Rotation 90^{0}

Beginning with ΔABC, draw the counterclockwise rotation of 90^{0 }centred at the origin. To 'observe' that this rotation is of 90^{0}, suppose point B is on the red arrow. The arrow is then turned 90^{0}(identify the 90^{0 }angle shown by the two red arrows). After that, look at point B's new position, which is labelled as B'. The same method of rotation can be used for all the three vertices.

Result: Rotation of 90^{0 }on Coordinate Axes

Centered at Origin

(x, y) -> (-y, x)

Rotation 180^{0}

Just like the previous case - beginning with ΔABC, draw the counterclockwise rotation of 180^{0}

centred at the origin. Again, as we did in the case of 90^{0}, suppose that point B is attached to the red arrow from the centre marking (0,0). Now, the arrow is moved 180^{0 }forming a straight line. Identify the new position of point B that is marked as B'.

Result: Rotation of Coordinate Axes

Centered at Origin

(x, y) -> (-x,-y)

Rotation 270^{0}

Beginning with quadrilateral ABCD, draw the counterclockwise rotation of 270^{0 }centred at the origin. Just like the previous two examples, suppose point A is attached to the arrow in red from the centre (0,0). The arrow is now moved 270^{0 }counterclockwise. Notice the new position of A, marked A'. Note that A' will also be "on" the axis as A was or is "on" the axis.

Result: Rotation of 270^{0 }on Coordinate Axes

Centred at origin

(x, y) → (y, -x )

From the above cases, we can conclude that rotation points towards an object's circular movement around a centre and that it's possible to rotate objects with different shapes by an angle around the centre. Moreover, we can rotate an object about an infinite number of imaginary lines, i.e., rotational axes. The rotations around x,y, and z axes are termed as the principle rotations, and they can be performed around any axis by taking the rotation around the x-axis, followed by the y-axis, and finally, the z-axis. This information can also help us in determining and understanding rotation formulas.

Rotation Formula

Rotation can undoubtedly be performed in both directions - clockwise and counterclockwise. The most common angles around which the rotation formula revolves are 90^{0}, 180^{0 }and 270^{0}.

Following these, certain rules for rotation in coordinate planes are:

As per Euclidean geometry, a rotation matrix refers to a matrix used to carry out a rotation in a Euclidean space. Let's understand this by taking an example of a two-dimensional cartesian plane approach, where the matrix R turns the points in the XY-plane in the counterclockwise direction by an angle θ about the origin. In this case, the matrix R will be provided by:

R = [cosΘsinΘ − sinΘcosΘ]

For performing the rotation by using the rotation matrix R, the position of every point in the plane is shown by a column vector “v," which contains the coordinate point. And, the rotated vector can finally be obtained by multiplying matrix Rv.

In geometry, there are a lot of shapes like circles, rectangles, and squares that have rotational symmetry. Furthermore, we can say that all the regular polygons are available with the aspect of rotational symmetry. A figure is said to have rotational symmetry if it appears exactly similar to how it was before getting rotated around its centre. Nevertheless, if we talk about the order of symmetry, then it can be calculated by counting the number of times the objects coincide with themselves while rotating through 360^{0}s.

Rotation Examples

The above-given example shows the turn, i.e. rotation of a rectangle by 90^{0 }each time. We can see that when the rectangle is rotated twice, its original shape is obtained at 180^{0}. Moreover, if it is rotated again twice, then its original shape will be obtained at 360^{0}. Hence, we can say that the rectangles have the rotational symmetry of order 2. In other words, the order of rotational symmetry of rectangles is 2.