Description
Check one basis/dual-vector dot constraint against zero or one under an absolute tolerance.
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.
Check one basis/dual-vector dot constraint against zero or one under an absolute tolerance.
When to use this tool
Use Vector Dual Basis One Dot Constraint 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 basis, dual, sameIndex, tolerance. 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 dot, residual, satisfies. Inspect satisfies before interpreting the associated values.
Require aligned basis and dual vectors and nonnegative tolerance. Compare their dot product with one when sameIndex is true or zero otherwise; return residual and satisfies under an inclusive absolute tolerance.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| basis | Basis | Required. Array of Finite number.; maximum items 256. |
| dual | Dual | Required. Array of Finite number.; maximum items 256. |
| sameIndex | Same Index | Required. Boolean. |
| tolerance | Nonnegative absolute tolerance in the quantity tested; see the stated units and boundary rule. | Required. Finite number; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| dot | Dot | Required. Finite number. |
| residual | Residual | Required. Finite number. |
| satisfies | Satisfies | Required. Boolean. |
{
"input": {
"basis": [
2,
0
],
"dual": [
0.5,
0
],
"sameIndex": true,
"tolerance": 0
},
"output": {
"dot": 1,
"residual": 0,
"satisfies": true
}
}Limitations and assumptions
- This is one duality constraint, not construction or verification of a complete dual basis. Empty arrays can satisfy a different-index zero constraint without establishing basis independence.
- 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
- Basis Vector Determinant Sign
Inspect floating-point orientation of a supplied 3D basis after row scaling; no exact-predicate guarantee.
- Vector Basis Coordinate Unit Scale
Rescale coordinates for independently scaled basis vectors by dividing by each supplied basis scale.