Rotate Vector Calculator

Example Data Table

Formula Used

2D Rotation Formula

For a point rotated around pivot (px, py), translate first:

tx = x - px, ty = y - py

Then rotate:

x' = tx cos θ - ty sin θ + px

y' = tx sin θ + ty cos θ + py

3D Rotation Formula

The 3D mode uses Rodrigues’ rotation formula:

v' = v cos θ + (k × v) sin θ + k(k · v)(1 - cos θ)

Here, k is the normalized axis vector. The vector is first moved relative to the selected center.

How To Use This Calculator

1. Select 2D or 3D rotation mode.
2. Enter vector components in the matching input fields.
3. Enter the rotation angle.
4. Choose degrees or radians.
5. Choose clockwise or counterclockwise rotation.
6. For 2D, enter a pivot point when needed.
7. For 3D, choose an axis or enter a custom axis.
8. Click Calculate Rotation.
9. Review the result above the form.
10. Download CSV or PDF when you need a copy.

What Is Vector Rotation?

Vector rotation changes a vector direction while keeping its length about a chosen center. It is useful in graphics, robotics, surveying, physics, mapping, and game logic. A point can spin around the origin. It can also spin around a pivot. In three dimensions, the rotation can happen around a selected axis.

Why This Calculator Helps

Manual rotation can create sign errors. Angles, units, clockwise direction, and pivot offsets all matter. This calculator keeps those steps together. It accepts degree or radian input. It supports 2D vectors, standard 3D axes, and custom 3D axes. It also reports the matrix values, translated vector, final components, and magnitude checks.

Important Input Choices

For 2D rotation, enter x and y. Then choose an angle and direction. Add a pivot when the vector should rotate around a point other than zero. The tool first subtracts the pivot. It then applies the rotation matrix. Finally, it adds the pivot back.

For 3D rotation, enter x, y, and z. Pick an axis, or enter a custom axis vector. A custom axis is normalized before use. The center point works like a 3D pivot. It lets a point rotate around an axis line passing through that center.

Reading The Result

The rotated components show the new vector or point location. The matrix section shows the actual coefficients used. Magnitude around the rotation center should stay almost unchanged. Very tiny differences may appear because computers round decimal values. The angle output helps you compare orientation before and after rotation.

Practical Uses

Designers use vector rotation to place objects at precise angles. Students use it to check matrix homework. Engineers use it for coordinate transformations. Developers use it for canvas movement, collision systems, camera control, and object animation. Survey and mapping work also depends on clean coordinate rotation.

Best Practices

Use consistent units. Check clockwise and counterclockwise settings before saving results. Use enough decimal places for technical work. When using a custom 3D axis, avoid a zero length axis. Export the report when you need to document the calculation or compare several scenarios.

Keep a small example set for checking future edits. It protects formulas when page changes are made. Inputs update safely during future revisions.

FAQs