Rotation Formula

Q: What do the symbols mean in the Rotation formula?

$R_{\theta}$ denotes rotation by angle $\theta$ about the origin (counterclockwise is positive)

The Formula

90° \text{ CCW}: (x, y) \to (-y, x) 180°: (x, y) \to (-x, -y) 270° \text{ CCW}: (x, y) \to (y, -x)

When to use: Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.

Quick Example

Rotate 90° counterclockwise about origin: (1, 0) \to (0, 1)

Notation

R_{\theta} denotes rotation by angle \theta about the origin (counterclockwise is positive)

What This Formula Means

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.

Formal View

R_\theta: \mathbb{R}^2 \to \mathbb{R}^2, R_\theta(x, y) = (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta); matrix form: R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, \det(R_\theta) = 1

Worked Examples

Example 1

easy

Rotate the point A(3, 2) by 90° counterclockwise about the origin. Where does A end up?

Solution

1
Step 1: The rule for 90° CCW rotation is (x, y) \to (-y, x).
2
Step 2: Apply to A(3, 2): A' = (-2, 3).
3
Step 3: Verify distance from origin is unchanged: \sqrt{3^2+2^2} = \sqrt{13} and \sqrt{(-2)^2+3^2} = \sqrt{13}. ✓

Answer

A' = (-2, 3)

The 90° CCW rotation rule (x,y)\to(-y,x) is derived from the rotation matrix with \theta=90°: \cos90°=0, \sin90°=1, giving (-y, x). Rotation preserves distance from the center of rotation.

Example 2

medium

Rotate the point B(4, 0) by 60° counterclockwise about the origin. Give exact coordinates.

Common Mistakes

Forgetting to specify the center of rotation — rotating around different centers gives different results
Confusing clockwise and counterclockwise direction — a 90° clockwise rotation is not the same as 90° counterclockwise
Thinking the center of rotation must be inside the shape — it can be any point

Why This Formula Matters

Models turning and spinning in physics, robotics, and engineering; forms the basis for circular motion.

Frequently Asked Questions

What is the Rotation formula?

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

How do you use the Rotation formula?

Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.