Rotation result
The result uses column vectors and the selected conventions.
Combined rotation matrix
—
Equivalent quaternion
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Equivalent Euler angles
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Formula used
Calculation steps
Batch rotation results
| # | Original | Rotated | Magnitude before | Magnitude after |
|---|
2D view
3D view
Calculation history
No saved calculations yet.
Formula used
Two-dimensional rotation
y′ = x sin(θ) + y cos(θ)
Three-dimensional axis rotation
Rodrigues’ arbitrary-axis formula
Quaternion rotation
How to use this calculator
- Select a rotation method and an input representation.
- Enter the vector, point, pivot, angle, or orientation values.
- Choose axis conventions, precision, and graph settings.
- Add sequence steps when several rotations are required.
- Press Calculate rotation to view all results and steps.
Example data
| Method | Input | Rotation | Expected result |
|---|---|---|---|
| 2D | (1, 0) | 90° | (0, 1) |
| Z-axis | (1, 0, 0) | 90° | (0, 1, 0) |
| X-axis | (0, 1, 0) | 90° | (0, 0, 1) |
| Arbitrary axis | (1, 0, 0) | 120° about (1,1,1) | (0, 1, 0) |
| Quaternion | (1, 0, 0) | (0.7071,0,0,0.7071) | (0, 1, 0) |
Frequently asked questions
1. Does rotation change vector magnitude?
A proper rotation preserves magnitude. Small differences can appear from rounding.
2. Why does rotation order matter?
Three-dimensional rotations usually do not commute. Changing order changes orientation.
3. What is an active rotation?
An active rotation moves the vector. The coordinate axes stay fixed.
4. What is a passive rotation?
A passive rotation changes coordinates between frames. It uses the inverse matrix.
5. Must an arbitrary axis be normalised?
Yes, Rodrigues’ formula requires a unit axis. Automatic normalisation is available.
6. Why must a quaternion be unit length?
A unit quaternion represents pure rotation. Other lengths introduce unwanted scaling.
7. What causes gimbal lock?
Two Euler axes can align at certain angles. One rotational degree becomes ambiguous.
8. Can I rotate around a custom pivot?
Yes. Select Point around a pivot and enter pivot coordinates.
9. How is a custom matrix validated?
The calculator checks dimensions, orthogonality, and determinant. Tolerance controls sensitivity.