Advanced Rotation Matrix Calculator

Rotation Matrix Calculator Form

Choose 2D or 3D mode, then calculate a standard axis rotation or a full Euler sequence.

Dimension

Angle Unit

3D Method

2D Direction

Axis

Euler Sequence

Angles A, B, and C follow the listed order.

Angle A

Angle B

Angle C

Vector X

Vector Y

Vector Z

Decimal Precision

Reset

Plotly Graph

The chart compares the original vector with the rotated vector in either 2D or 3D space.

Example Data Table

Case	Mode	Setup	Input Vector	Rotated Vector
1	2D	30 deg CCW	(4, 2)	(2.464, 3.732)
2	2D	90 deg CCW	(3, 1)	(-1, 3)
3	3D	X axis, 45 deg	(3, 2, 1)	(3, 0.707, 2.121)
4	3D	Y axis, 90 deg	(2, 0, 1)	(1, 0, -2)

Formula Used

2D Rotation Matrix

For a planar angle θ, the rotation matrix is:
R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]]

For an input vector v = [x, y], the rotated vector is:
v′ = R(θ) · v

3D Axis Rotations

For 3D work, this calculator uses standard right-hand-rule matrices:

Rx(θ) rotates around the X axis.
Ry(θ) rotates around the Y axis.
Rz(θ) rotates around the Z axis.

Euler Sequence Composition

If the sequence is, for example, ZYX, the calculator applies the first listed rotation, then the second, then the third, using matrix multiplication to produce one combined rotation matrix.

Useful Properties

Pure rotation matrices are orthogonal.
The inverse equals the transpose.
The determinant should be +1.
Vector magnitude is preserved by exact rotations.

How to Use This Calculator

Select 2D or 3D mode.
Choose degrees or radians.
For 3D, choose a single axis or an Euler sequence.
Enter one, two, or three angles as needed.
Enter the vector coordinates you want to rotate.
Choose your preferred output precision.
Press Calculate Rotation Matrix.
Review the result block shown above the form.
Download the results as CSV or PDF if needed.

FAQs

1. What does a rotation matrix do?

A rotation matrix changes a vector’s direction without changing its length. It re-expresses coordinates after a rotation in 2D or 3D space while preserving rigid geometry.

2. Why is the determinant usually 1?

A pure rotation preserves area or volume and does not reflect the object. That is why the determinant of a valid rotation matrix is expected to be positive one.

3. Why is the inverse equal to the transpose?

Rotation matrices are orthogonal. Their columns remain unit-length and mutually perpendicular, so the transpose exactly reverses the rotation and becomes the inverse.

4. What is the difference between single-axis and Euler rotation?

Single-axis rotation turns the vector around one chosen axis only. Euler rotation combines three ordered axis rotations to build a more general spatial orientation change.

5. Does rotation change vector magnitude?

No. Ideal rotation keeps vector magnitude unchanged. Small differences can appear only from rounding in displayed output, not from the underlying rotation concept itself.

6. Why does Euler order matter?

Matrix multiplication is not commutative in general. Rotating around X, then Y, then Z usually produces a different orientation than performing those same angles in another order.

7. When should I use radians instead of degrees?

Use radians when working directly with mathematical formulas, programming libraries, or calculus-based derivations. Use degrees when you want more familiar engineering or classroom input values.

8. Can this calculator help verify matrix quality?

Yes. It reports determinant, transpose, inverse behavior, and orthogonality error. These checks help confirm whether the computed matrix behaves like a proper rotation matrix.