Rotation

The term rotation is common in Maths as well as in science. It is applicable for the rotational or circular motion of some object around the centre or some axis. In our real-life, we all know that earth rotates on its own axis, which is a natural rotational motion. In geometry, four basic types of transformations are Rotation, Reflection, Translation, and Resizing. This article will give the very fundamental concept about the Rotation and its related terms and rules. Let us begin it!

Rotation

Source: en.wikipedia.org

What is Rotatory Motion?

The motion of some rigid body which takes place so that all of its particles move in the circles about an axis with a common velocity. Also, the rotation of the body about the fixed point in the space. Such motions are also termed as rotational motion.

Rotational motion is more complex in comparison to linear motion. But, many of the equations for the mechanics of the rotating body are similar to the linear motion equations. Thus, in Physics, the general laws of motions are also applicable for the rotational motions with their equations.

Rotation Science Definition:

Rotation means the circular movement of somebody around a given centre. It is possible to rotate many shapes by the angle around the centre point. In three-dimensional shapes, the objects can rotate about an infinite number of imaginary lines known as rotation axis or axis of motion.

The rotations around the X, Y and Z axes are termed as the principal rotations. The point about which the object rotates is the rotation about a point.

The rotation transformation is about turning a figure along with the given point. In the mathematical term rotation axis in two dimensions is a mapping from the XY-Cartesian point system. Thus A rotation is a transformation in which the body is rotated about a fixed point. The direction of rotation may be clockwise or anticlockwise.

The point about which the object is rotating, maybe inside the object or anywhere outside it. The amount of rotation is in terms of the angle of rotation and is measured in degrees. As a convention, we denote the anti-clockwise rotation as a positive angle and clockwise rotation as a negative angle.

Rotation Formula:

Rotation can be done in both directions like clockwise and anti-clockwise. Common rotation angles are \(90^{0}\), \(180^{0}\) and \(270^{0}\) degrees. There are rotation rules for rotation in the coordinate plane at these angles. Rules for point (x,y) are:

Rotation	Point  coordinate	Point coordinate after Rotation
Rotation of \(90^{0}\) (Clockwise)	(x, y)	(y, -x)
Rotation of \(90^{0}\) (Anti-Clockwise)	(x, y)	(-y, x)
Rotation of \(180^{0}\) (Both)	(x, y)	(-x, -y)
Rotation of \(270^{0}\) (Clockwise)	(x, y)	(-y, x)
Rotation of \(270^{0}\) (Anti-Clockwise)	(x, y)	(y, -x)

Solved examples for You

Q.1: What is the rotational symmetry?

Solutions: In geometry, many shapes are having rotational symmetry such as circle, square, rectangle. Also, all the regular polygons are having rotational symmetry. If an object is rotated around the centre point, the object appears exactly the same as before the rotation. Then such objects are said to have rotational symmetry.

Q.2: What will be the coordinate of a point having coordinate (3,-6) after rotations as:

(i)         \(90^{0}\) clock-wise

(ii)        \(270^{0}\) anti-clockwise

(iii)       \(90^{0}\) anti-clockwise

(iv)       \(270^{0}\) clockwise.

Solution: Coordinates will be given as, based on the rotation formula and transformation rules:

(v)        \(90^{0}\) clock-wise    : (-6,-3)

(vi)       \(270^{0}\) anti-clockwise : (-6.-3)

(vii)      \(90^{0}\) anti-clockwise : (6,3)

(viii)     \(270^{0}\) clockwise : (6,3)