Calculator Inputs
The page uses stacked sections, while the calculator fields adapt to 3 columns on large screens, 2 on medium screens, and 1 on small screens.
Example data table
| Case | Input | Rotation setup | Expected rotated result |
|---|---|---|---|
| 2D origin case | (3, 4) | 90° counterclockwise about (0, 0) | (-4, 3) |
| 2D pivot case | (5, 2) | 180° about pivot (1, 1) | (-3, 0) |
| 3D principal axis | (1, 2, 3) | 90° about Z-axis | (-2, 1, 3) |
| 3D arbitrary axis | (1, 0, 0) | 120° about axis (1, 1, 1) | (0, 1, 0) |
Formula used
1) 2D rotation about a pivot
Translate the point to the pivot, rotate it with the 2D matrix, then translate it back.
2) 3D principal-axis rotation
The calculator applies standard right-hand-rule matrices for X, Y, or Z rotations.
3) Arbitrary-axis rotation
For axis vector k = (u, v, w), the calculator normalizes the axis and applies Rodrigues’ formula.
4) Euler sequence rotation
For a chosen order such as XYZ, the tool multiplies the step matrices sequentially. Matrix order matters, so XYZ and ZYX usually produce different results.
How to use this calculator
- Choose 2D or 3D mode.
- Enter the vector components.
- Select angle units in degrees or radians.
- For 2D, set the pivot and direction.
- For 3D, choose principal axis, arbitrary axis, or Euler sequence.
- Set the precision you want in the output.
- Press Rotate Vector.
- Review the rotated vector, displacement, matrix, and graph above the form.
- Use the CSV or PDF buttons to export the result summary.
Frequently asked questions
1) What does vector rotation mean?
Vector rotation changes direction while following a chosen angle and axis. In pure origin-based rotation, the length stays unchanged because the transformation matrix is orthogonal.
2) Does rotation always preserve magnitude?
Yes for standard vector rotation about the origin in 2D or 3D. If you rotate a point around another pivot, the distance from that pivot stays fixed, but the distance from the origin may change.
3) Why do Euler orders give different answers?
3D rotations are not generally commutative. Rotating around X, then Y, then Z is usually different from rotating around Z, then Y, then X.
4) When should I use arbitrary-axis rotation?
Use it when the rotation axis is not aligned with the global X, Y, or Z axes. This is common in rigid-body motion, robotics, graphics, and spatial geometry problems.
5) What is the determinant telling me?
A proper rotation matrix should have determinant 1. That indicates orientation is preserved and the matrix represents rotation without reflection or scaling.
6) What is orthogonality error?
It measures how close the matrix is to an ideal rotation matrix. Smaller values mean the matrix columns remain perpendicular and unit-length, as expected for rotation.
7) Can I use negative angles?
Yes. Negative angles rotate in the opposite direction under the chosen convention. In 3D, the calculator follows the right-hand rule for positive angles.
8) What do the export buttons include?
They export the current result summary, including method, vectors, displacement, magnitudes, determinant, orthogonality error, and the rotation matrix in flattened text form.