In this lesson, we delve into the concept of rotation, a fundamental transformation in both mathematics and coordinate geometry. Rotation refers to moving a figure or a point around a fixed center in a circular path. It is characterized by three main factors: the center of rotation, the angle of rotation, and the direction of rotation (clockwise or counterclockwise).
Center of Rotation: This is a fixed point around which the rotation occurs. It can be a point within the figure, outside it, or at one of its vertices.
Angle of Rotation: This is the measure of rotation, in degrees or radians, indicating the extent of rotation. A positive angle denotes counterclockwise rotation, while a negative angle indicates clockwise rotation.
Direction of Rotation: Rotations can be performed in two directions - clockwise or counterclockwise.
In coordinate geometry, when we rotate a point or an object, its position changes according to specific rules depending on the angle of rotation. Here are the rules for rotating points around the origin (0,0) in the coordinate plane:
Rotating points around any other center, \(C(h, k)\), requires adjusting the positions before and after the rotation to account for the shift in origin.
The mathematical representation of rotation uses rotation matrices. A rotation matrix can rotate points in the plane around the origin through an angle \(\theta\). For counterclockwise rotation, the rotation matrix is:
\( R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \)To rotate a point \(P(x, y)\) around the origin by an angle \(\theta\), we multiply its coordinates by the rotation matrix:
\( P'(x', y') = R(\theta) \cdot P(x, y) \)This operation transforms the original coordinates \((x, y)\) into the new coordinates \((x', y')\) after the rotation.
Example 1: Consider a point \(P(2, 3)\) on the Cartesian plane. To rotate this point 90° counterclockwise about the origin, we apply the formula for a 90° counterclockwise rotation, yielding the new position \(P'(3, -2)\).
Example 2: If we have a rectangle with corners at \(A(1, 1)\), \(B(1, 4)\), \(C(5, 4)\), and \(D(5, 1)\), and we want to rotate this rectangle 180° around the origin, each point's new position will be \(A'(-1, -1)\), \(B'(-1, -4)\), \(C'(-5, -4)\), and \(D'(-5, -1)\).
Rotation is not just a mathematical concept but also a real-world phenomenon. For instance, the Earth rotates around its axis, resulting in day and night. Similarly, wheel rotations allow vehicles to move. In sports, athletes use rotation techniques to enhance their performance in activities such as discus throw or figure skating.
One simple experiment to understand rotation involves using a piece of paper and a pencil. Draw a shape with well-defined vertices on the paper. Pin the paper at a point that will serve as the center of rotation. Using the pencil, trace the path of each vertex as you rotate the paper by a specific angle. The traced points mark the new positions of the shape's vertices after rotation.
Understanding rotation helps in numerous areas beyond mathematics and geometry, including physics, engineering, computer graphics, and robotics. It is a key concept in designing and interpreting the movement and orientation of objects in two and three-dimensional spaces.
