1. What is a Rotation Matrix?

A rotation matrix is a linear transformation matrix that performs a rotation in Euclidean space. For 2D space, it rotates points in the plane around the origin (0,0) by a specified angle.

2. How Does the Calculator Work?

The calculator computes the 2D rotation matrix:

Where:

• \( \theta \) — Rotation angle in radians
• \( \cos\theta \) — Cosine of the angle
• \( \sin\theta \) — Sine of the angle

Explanation: The matrix transforms point coordinates (x,y) to new coordinates (x',y') by rotating them θ radians counterclockwise about the origin.

3. Applications of Rotation Matrices

Details: Rotation matrices are fundamental in computer graphics, robotics, physics simulations, and any application involving coordinate transformations.

4. Using the Calculator

Tips: Enter the rotation angle in radians (π radians = 180°). The calculator will compute and display the 2×2 rotation matrix with 4 decimal precision.

5. Frequently Asked Questions (FAQ)

Q1: How do I convert degrees to radians?
A: Multiply degrees by π/180 (approximately 0.0174533). For example, 90° = 90 × π/180 = π/2 radians.

Q2: Does the rotation preserve distances?
A: Yes, rotation matrices are orthogonal and preserve vector lengths and angles between vectors.

Q3: What's the difference between 2D and 3D rotation matrices?
A: 3D rotations are more complex, requiring separate matrices for x, y, and z axis rotations or using Euler angles/quaternions.

Q4: Can I combine multiple rotations?
A: Yes, by multiplying the matrices (note: matrix multiplication is not commutative - order matters).

Q5: What's the inverse of a rotation matrix?
A: The inverse is its transpose, which corresponds to rotating by the negative angle.