1. What is Vector Rotation?

Vector rotation is a transformation that changes the direction of a vector while maintaining its magnitude. It's a fundamental operation in computer graphics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses the standard rotation formula:

Where:

• \( x \) — Original x-coordinate
• \( y \) — Original y-coordinate
• \( \cosθ \) — Cosine of rotation angle
• \( \sinθ \) — Sine of rotation angle

Explanation: The formula rotates a point (x,y) around the origin by angle θ in the counter-clockwise direction.

3. Importance of Vector Rotation

Details: Vector rotation is essential in computer graphics for object transformations, in physics for coordinate system transformations, and in robotics for motion planning.

4. Using the Calculator

Tips: Enter the original x and y coordinates, then specify the rotation angle in degrees. Positive angles rotate counter-clockwise, negative angles rotate clockwise.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between 2D and 3D rotation?
A: 2D rotation occurs around a single point (the origin), while 3D rotation occurs around an axis and requires more complex matrix operations.

Q2: How does rotation affect vector magnitude?
A: Pure rotation maintains the original vector magnitude (length), only changing its direction.

Q3: What's the rotation matrix?
A: The 2D rotation matrix is \[ \begin{bmatrix} \cosθ & -\sinθ \\ \sinθ & \cosθ \end{bmatrix} \] which transforms the original coordinates.

Q4: Can I rotate around an arbitrary point?
A: To rotate around a point (a,b), first translate by (-a,-b), rotate, then translate back by (a,b).

Q5: How are rotation angles typically measured?
A: In mathematics, positive angles are counter-clockwise from the positive x-axis, while negative angles are clockwise.