Online Rotation Matrix Calculator

Calculator Input

Example Data Table

Formula Used

2D rotation:
R = [[cos θ, -sin θ],
     [sin θ,  cos θ]]

3D x-axis rotation:
Rx = [[1, 0, 0],
      [0, cos θ, -sin θ],
      [0, sin θ,  cos θ]]

3D y-axis rotation:
Ry = [[ cos θ, 0, sin θ],
      [0, 1, 0],
      [-sin θ, 0, cos θ]]

3D z-axis rotation:
Rz = [[cos θ, -sin θ, 0],
      [sin θ,  cos θ, 0],
      [0, 0, 1]]

Custom axis rotation:
R = I cos θ + (1 - cos θ)uuᵀ + [u]x sin θ

Vector transformation:
v' = Rv

Rotation matrix checks:
det(R) ≈ 1
R⁻¹ = Rᵀ
RᵀR ≈ I

How To Use This Calculator

1. Select the rotation mode. Use 2D for planar points.
2. Enter the angle and choose degrees or radians.
3. Enter the vector coordinates. Use z for spatial rotations.
4. For custom axis mode, enter the axis vector components.
5. For Euler mode, enter x, y, and z angles.
6. Choose precision and press the calculate button.
7. Review the matrix, transformed vector, determinant, and graph.
8. Use CSV or PDF download for reports and notes.

Understanding Rotation Matrices

A rotation matrix turns a point, vector, or coordinate frame without changing its length. It is used in geometry, robotics, graphics, mapping, games, physics, and classroom work. The idea is simple. The matrix stores cosines and sines. They describe how each axis turns.

Why This Calculator Helps

Manual rotation work can be slow. A sign error can move a point in the wrong direction. This calculator builds the matrix from the angle and method. It then multiplies the matrix by the chosen vector. It also checks the determinant and orthogonality. Those checks show whether the matrix behaves like a true rotation.

2D And 3D Use Cases

In two dimensions, the matrix rotates points around the origin. A positive angle usually means a counterclockwise turn. In three dimensions, rotation can happen around the x axis, y axis, z axis, or a custom axis. Custom axis rotation uses a normalized axis vector. That method helps when the turn is not aligned with a main coordinate axis.

Reading The Results

The transformed vector shows the new point or direction. The determinant should be close to one. The inverse should match the transpose for a true rotation matrix. Small decimal differences are normal. Computers round trigonometric values. The graph gives a quick visual check. It compares the original and rotated vector in the xy plane.

Common Input Choices

For planar work, select 2D mode and enter x and y. For a standard spatial turn, choose one main axis. For advanced spatial work, enter a custom axis. The calculator normalizes that axis before forming the matrix. Euler mode helps with chained turns. It applies three angles in a selected order. Order matters greatly.

Best Practices

Use degrees when working from diagrams. Use radians when copying formulas from textbooks. Keep enough precision for engineering work. Test simple cases first. A 90 degree z axis rotation should move the x direction toward y. Export the table when you need to document results. Use the formula section to verify each step.

Accuracy Notes

Angles near zero should return the original vector. Large angles work because sine and cosine repeat. Rounding affects display, not stored calculations.

FAQs