Description
Calculate a scaled normalized dot product and its residual from a supplied cosine.
A vector is an ordered set of components in a chosen basis. Its norm measures magnitude, while dot products, projections, and coordinate transformations describe different relationships. Component order, angle units, normalization, and numerical tolerances determine the meaning of a computed result.
Calculate a scaled normalized dot product and its residual from a supplied cosine.
When to use this tool
Use Vector Normalized Dot Residual when you need the particular coordinate operation, derivative, residual, or domain check under a known basis convention.
How to use the tool and how it works
- Enter first, second, suppliedCosine. Keep parallel arrays aligned. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read cosine, residual. Use the output convention below.
Require matching nonzero vectors and suppliedCosine within minus one through one. Scale each vector by its maximum absolute component, normalize it, then dot the normalized vectors for cosine and return residual=suppliedCosine-cosine.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| first | First | Required. Array of Finite number.; maximum items 256. |
| second | Second | Required. Array of Finite number.; maximum items 256. |
| suppliedCosine | Supplied Cosine | Required. Finite number; minimum -1. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| cosine | Cosine | Required. Finite number. |
| residual | Residual | Required. Finite number. |
{
"input": {
"first": [
3,
0
],
"second": [
0,
4
],
"suppliedCosine": 0.2
},
"output": {
"cosine": 0,
"residual": 0.2
}
}Limitations and assumptions
- Scaling reduces overflow risk during normalization, but the calculation remains floating point and does not clamp cosine or use an acceptance tolerance. Residual compares a supplied target rather than validating an angle measurement.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Choose a complete vector or matrix operation when a single component or residual is insufficient. For sensitive near-zero or near-singular cases, inspect numerical conditioning and use an appropriate precision and tolerance.
References
Similar or alternative tools
- Vector 2d Shear Area Preservation Residual
Compare oriented planar determinants before and after a supplied x-by-y shear.
- Vector Affine Combination Weight Residual
Sum weights and return residual=sum-1.