Rotation in the Cartesian Plane

What is Rotation?

Rotation is a type of transformation that turns a shape around a fixed point, known as the center of rotation. In the Cartesian plane, this point is often the origin \$(0, 0) \\$.

A rotation is defined by:

• Angle of Rotation: The degree through which the shape is rotated.
• Direction: Either clockwise or counterclockwise.

Rotation About the Origin

When rotating a point \$(x, y) \\$ about the origin by an angle \$\theta \\$, the new coordinates \$(x', y') \\$ are given by:

• Clockwise Rotation: \$$x' = x \cos \theta + y \sin \theta \\$$ \$$y' = -x \sin \theta + y \cos \theta \\$$
• Counterclockwise Rotation: \$$x' = x \cos \theta - y \sin \theta \\$$ \$$y' = x \sin \theta + y \cos \theta \\$$

Common Rotation Angles

Angle	Clockwise Transformation	Counterclockwise Transformation
90°	$(x, y) \rightarrow (y, -x)$	$(x, y) \rightarrow (-y, x)$
180°	$(x, y) \rightarrow (-x, -y)$
270°	$(x, y) \rightarrow (-y, x)$	$(x, y) \rightarrow (y, -x)$