Enter Rotation Inputs
Use the elimination pair to create the rotation. The visualization vector and local block help you inspect the same rotation from multiple angles.
Example Data Table
This sample demonstrates a common elimination step where the lower entry becomes zero after rotation.
| a | b | c | s | r | Angle ° | Rotated Pair | Updated Block Row 1 | Updated Block Row 2 |
|---|---|---|---|---|---|---|---|---|
| 6 | 8 | 0.600000 | -0.800000 | 10.000000 | -53.130102 | (10.000000, 0.000000) | (10.000000, 5.200000) | (0.000000, 1.400000) |
Formula Used
r = √(a² + b²) or numerically r = hypot(a, b)
c = a / rs = -b / rWhen
r = 0, use c = 1 and s = 0.
G = [ [c, -s], [s, c] ]
G[a; b] = [ c·a - s·b ; s·a + c·b ] = [ r ; 0 ]
For a local block
[ [m11, m12], [m21, m22] ], left multiplication gives:new11 = c·m11 - s·m21new12 = c·m12 - s·m22new21 = s·m11 + c·m21new22 = s·m12 + c·m22
How to Use This Calculator
- Enter the elimination pair a and b. The tool builds a rotation that removes the lower value.
- Optionally enter a separate vector (x, y) to see how the same rotation transforms another point.
- Fill the 2 × 2 local block if you want to inspect how neighboring matrix columns change during a QR-style step.
- Choose the output precision that matches your study or reporting needs.
- Press Calculate Rotation to show results above the form, directly under the header area.
- Review the matrix, rotated vectors, norm preservation, and residual zeroing check.
- Use the CSV and PDF buttons to export the summary table for notes, homework, or documentation.
Frequently Asked Questions
1) What does a Givens rotation do?
It rotates a two-component vector so one chosen entry becomes zero. This makes it useful for QR factorization, least-squares solvers, and structured elimination steps.
2) Why is the value r important?
The quantity r is the length of the pair (a, b). It becomes the remaining top entry after rotation and helps define the cosine and sine terms.
3) Why use hypot instead of direct square roots?
The hypot function is numerically safer for very large or very small values. It reduces overflow and underflow risk while computing the same Euclidean length.
4) Does the rotation preserve vector length?
Yes. A proper Givens rotation is orthogonal, so it preserves norms and angles. The calculator reports a norm difference to verify this behavior numerically.
5) Why can the angle be negative?
The sign of the sine term determines rotation direction. A negative angle means the vector turns clockwise under the chosen convention.
6) Where is this method used in practice?
It appears in numerical linear algebra, QR decomposition, least-squares fitting, signal processing, and sparse matrix algorithms where local transformations are preferred.
7) What happens if both a and b are zero?
No rotation is needed. The calculator returns the identity rotation, meaning c = 1 and s = 0, because every direction is already valid.
8) Why does the calculator update a 2 × 2 block?
That block shows how the same row rotation affects nearby columns during elimination. It helps connect the two-entry theory to real matrix updates.