Rotation

Rotation is a type of rigid body transformation which changes the orientation of the object. The most common type of rotation is a rotation around the origin. To look at how rotations are performed, let’s take Point A, which is located at the coordinates of (a, b), as shown below:

Rotation 1.png

Using the next image, the length of the yellow line is the value of a, and the length of the green line is the value of b.

Rotation 2

To rotate the yellow and green lines, 90º around the origin gives us the following situation:

Rotation 3.png

To find the coordinates of where the image, point A’, we see that the green line is now in the negative x-direction, and the yellow line is now in the y-direction. So, the coordinates of A’ would be (-b, a).

Rotation 4.png

This is the way to perform any 90º counterclockwise around the origin, you switch the values of the and coordinate values and make the new x value negative. If we wanted to perform a 180º counterclockwise rotation on around the origin on point A, would be the same as doing a 90º counterclockwise rotation point A’. So, the coordinates of this new point would be (-a, -b).

Rotation 5.png

To perform a 270º counterclockwise rotation around the origin would move the point to coordinates (b, -a).

Rotation 6.png

Use the table below to help you determine the coordinates of a point after a rotation was performed:

Rotation Amount Direction Coordinates
Counterclockwise (a, b)
90º  Counterclockwise (-b, a)
180º  Counterclockwise (-a, -b)
270º  Counterclockwise (b, -a)
360º  Counterclockwise (a, b)
90º  Clockwise (b, -a)
180º  Clockwise (-a, -b)
270º  Clockwise (-b, a)
360º  Clockwise (a, b)