Description
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; rotate a finite three-vector by a validated unit quaternion without applying translation.
A quaternion has one scalar and three vector components. This family uses [w, x, y, z] unless a layout converter says otherwise. Unit quaternions can represent rotations, and q and −q describe the same rotation under this convention; component arithmetic and physical rotation interpretation are separate operations.
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; rotate a finite three-vector by a validated unit quaternion without applying translation.
When to use this tool
Use this tool to rotate a finite three-vector by a validated unit quaternion without applying translation.
How to use the tool and how it works
- Enter quaternion, vector using the field definitions below.
- Run the operation. Correct input errors before interpreting an output.
- Read result.vector. Use the ordering, boundary rules and absence conditions described below.
Require a unit scalar-first quaternion and a three-component vector. Compute q × [0, vector] × conjugate(q) and return the vector part.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| quaternion | Four scalar-first components [w, x, y, z], unit length when required above. | Dense list of at most 128 finite numbers; integer/shape constraints above also apply. |
| vector | Three vector components in the coordinate convention being rotated. | Dense list of at most 128 finite numbers; integer/shape constraints above also apply. |
| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.vector | Three rotated vector components in the input coordinate convention. |
{
"input": {
"quaternion": [
0,
0,
0,
1
],
"vector": [
1,
2,
3
]
},
"output": {
"result": {
"vector": [
-1,
-2,
3
]
}
}
}Limitations and assumptions
- This rotates a vector without translation; coordinate-frame interpretation must match your application.
- When an operation requires unit input, the fixed acceptance condition is |norm − 1| ≤ 1e−10. The norm audit instead uses its supplied tolerance. Angles use radians; vector units are preserved by a unit rotation. Floating-point roundoff is possible.
- Every declared top-level input field is required and unexpected fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
- Numeric outputs must be finite and no larger in absolute magnitude than the safe-integer limit; operations returning exact large integers use decimal text. Positions are zero-based and spans end exclusively unless the operation explicitly states a different convention.
Alternative or Complementary methods
Confirm component layout, multiplication order and coordinate-frame convention before using another library’s records. Normalize only when appropriate, and distinguish component-space tolerances from angular tolerances.
References
Similar or alternative tools
- Unit Quaternion Basis Vector Images
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; return the three active basis-vector images as columns-as-vectors, not a transposed row-major matrix convention.
- Quaternion Unit Relative Rotation
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; compute the relative active rotation to * conjugate(from), normalized to remove numerical drift.