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Quaternion Arithmetic and Rotation Calculator — result sheet
Multiply two quaternions, inspect norms, conjugates, inverses, and rotate a 3D vector with a normalized quaternion.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: (w₁,v₁)(w₂,v₂) = (w₁w₂ − v₁·v₂, w₁v₂ + w₂v₁ + v₁×v₂); q⁻¹ = conjugate(q)/||q||²; v′ = q̂(0,v)q̂⁻¹.
A quaternion has one scalar component and a three-component vector part. This page keeps Hamilton multiplication and the rotation convention visible: the first quaternion is normalized before it rotates the entered vector, while the raw norms and inverse remain available for checking the arithmetic.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Quaternion 1 scalar w — minimum -1000000; maximum 1000000
- Quaternion 1 i component x — minimum -1000000; maximum 1000000
- Quaternion 1 j component y — minimum -1000000; maximum 1000000
- Quaternion 1 k component z — minimum -1000000; maximum 1000000
- Quaternion 2 scalar w — minimum -1000000; maximum 1000000
- Quaternion 2 i component x — minimum -1000000; maximum 1000000
- Quaternion 2 j component y — minimum -1000000; maximum 1000000
- Quaternion 2 k component z — minimum -1000000; maximum 1000000
- Vector x — minimum -1000000; maximum 1000000
- Vector y — minimum -1000000; maximum 1000000
- Vector z — minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Quaternion 1 scalar w | 0.7071067812 |
| Quaternion 1 i component x | 0 |
| Quaternion 1 j component y | 0 |
| Quaternion 1 k component z | 0.7071067812 |
| Quaternion 2 scalar w | 1 |
| Quaternion 2 i component x | 0 |
| Quaternion 2 j component y | 0 |
| Quaternion 2 k component z | 0 |
| Vector x | 1 |
| Vector y | 0 |
| Vector z | 0 |
The first quaternion is approximately a 90° rotation about z; it rotates (1, 0, 0) to approximately (0, 1, 0).
Assumptions and limits
- Quaternion components use the scalar-first convention q = w + xi + yj + zk and the Hamilton product order q1 × q2.
- The vector rotation uses the normalized first quaternion q̂ and the active relation q̂(0,v)q̂⁻¹.
- Both quaternions must be nonzero so normalization and inverse are defined.
- This is numerical quaternion arithmetic; it does not infer a coordinate frame, sensor convention, handedness, or physical attitude from unlabeled data.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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