Calculator Inputs
Example Data Table
| Case | Dimension | Angle | Axis | Input Vector | Expected Rotated Vector |
|---|---|---|---|---|---|
| Quarter turn | 2D | 90° | Origin | [1, 0] | [0, 1] |
| Half turn | 2D | 180° | Origin | [2, 3] | [-2, -3] |
| Z axis turn | 3D | 90° | [0, 0, 1] | [1, 0, 0] | [0, 1, 0] |
| X axis turn | 3D | 90° | [1, 0, 0] | [0, 1, 0] | [0, 0, 1] |
Formula Used
2D Rotation Matrix
For angle θ, the active counterclockwise rotation matrix is:
R = [[cos θ, -sin θ], [sin θ, cos θ]]
The rotated vector is:
[x', y'] = R × [x, y]
3D Rotation Matrix
For a normalized axis u = [ux, uy, uz], this calculator uses Rodrigues' rotation formula.
R = I cos θ + (1 - cos θ)uuᵀ + [u]× sin θ
Validation Checks
det(R) = 1 for a proper rotation.
R⁻¹ = Rᵀ because rotation matrices are orthogonal.
RᵀR = I is checked through the orthogonality error.
How To Use This Calculator
- Select 2D or 3D rotation.
- Enter the rotation angle.
- Choose degrees or radians.
- Select clockwise or counterclockwise direction.
- Choose active vector rotation or passive frame rotation.
- For 3D, select a standard axis or enter a custom axis.
- Enter the vector coordinates.
- Press calculate and review the result above the form.
- Use CSV or PDF export for reports and records.
Rotation Matrix Guide
Why Rotation Matrices Matter
A rotation matrix describes a turn without changing length. It preserves angles, area in two dimensions, and volume in three dimensions. This makes it useful in graphics, robotics, physics, navigation, and classroom geometry. A good calculator should show more than a final point. It should show the matrix, the rotated vector, the determinant, and the checks behind the answer.
Two Dimensional Rotation
In two dimensions, one angle controls the whole transformation. A positive angle usually means a counterclockwise turn. The matrix uses cosine and sine. When it multiplies a column vector, the vector moves around the origin. The determinant stays one. The inverse equals the transpose. These facts confirm that the operation is a pure rotation, not scaling or shearing.
Three Dimensional Rotation
In three dimensions, an angle is not enough. You also need an axis. The axis may be the x, y, or z direction. It may also be a custom vector. The calculator normalizes that vector first. Then it applies Rodrigues' formula. This creates a matrix that turns points around the chosen axis while keeping the axis itself fixed.
Accuracy And Interpretation
Small rounding errors are normal. They happen because computers store decimals with limited precision. The orthogonality error helps you judge accuracy. A value near zero means the rows and columns remain perpendicular. The determinant should be close to one. The trace can also help analyze the angle, especially when checking three dimensional results.
Practical Workflow
Enter the angle, choose the unit, select a direction, and provide the point. For three dimensional work, select the axis carefully. Use the graph to compare the original and rotated vectors. Export the result when you need records for homework, design notes, or technical reports. The example table gives ready tests for quick verification.
Common Checks
For a trusted result, compare lengths before and after rotation. They should match. Watch the sign of the angle. Clockwise and passive frame settings can reverse the visible motion. If a custom axis is entered, avoid a zero vector. A zero axis has no direction, so a stable three dimensional rotation cannot be formed. Then save exports safely.
FAQs
What is a rotation matrix?
A rotation matrix is a square matrix that turns vectors around an origin or axis. It preserves length and angle, so it represents pure rotation without scaling or distortion.
What is the difference between 2D and 3D rotation?
A 2D rotation needs one angle. A 3D rotation needs an angle and an axis. The axis tells the calculator which line the vector rotates around.
Why is the determinant close to one?
A proper rotation preserves area or volume. That property makes its determinant equal to one. Small decimal differences can appear because of rounding.
Why does the inverse equal the transpose?
Rotation matrices are orthogonal. Their rows and columns stay perpendicular and unit length. Because of this, the inverse matrix is the same as the transpose.
What is active rotation?
Active rotation turns the vector itself while the coordinate frame stays fixed. It is the common choice for moving points or objects in space.
What is passive rotation?
Passive rotation changes the coordinate frame instead of moving the vector. It often uses the opposite angle compared with active rotation.
Can I use a custom 3D axis?
Yes. Choose custom axis and enter x, y, and z components. The calculator normalizes the axis before applying Rodrigues' formula.
Why is my rotated vector length unchanged?
A rotation changes direction, not magnitude. If the matrix is correct, the original vector length and rotated vector length should match closely.