Source: https://guthlabs.ai/docs/instruments/vector-operations-toolkit/mechanism/

instrument mechanism / verified-clean

# Vector Operations Toolkit mechanism

The inspectable method record for Vector Operations Toolkit, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The supplied dot and projection formulas are mathematically correct under their constraints. The entry cannot receive PASS until the missing magnitude, normalization, and angle formulas and the exact 2D cross-product and angle-unit conventions are specified.

- The dot-product formula is standard, but it requires a and b to have the same dimension.

- The cross-product determinant form is standard for 3D vectors, but the specification does not define the 2D behavior. A 2D cross product may be represented as the scalar determinant a1b2-a2b1 or as a 3D vector with zero x and y components.

- Magnitude and normalization are claimed outputs but their formulas are not specified. Standard formulas are |a|=sqrt(Σai²) and normalize(a)=a/|a|.

- The projection formula is standard for projection of a onto b, but it requires |b|²≠0.

- The angle-between-vectors formula is missing. The standard formula is angle=acos((a·b)/(|a||b|)), with both vectors nonzero and the acos argument constrained to [-1,1].

- The specification claims 2D and 3D support but does not define whether mixed dimensions are rejected or padded.

## Units and conventions

- Vector components must use compatible units and the two vectors must have matching dimensions for dot, cross, projection, and angle operations.

- For ordinary geometric vectors, components are assumed to use the same unit within each vector. Magnitude and projection then have the component unit; dot products have squared units if the components are dimensional.

- The reported angle is assumed to be in radians unless the interface explicitly selects degrees. The specification does not state this unit.

- The 2D cross-product test below assumes the scalar determinant convention cross_z=a1b2-a2b1.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "a": [
      3,
      4,
      0
    ],
    "b": [
      1,
      2,
      2
    ]
  },
  "expected_outputs": {
    "dot": 11,
    "cross": [
      8,
      -6,
      2
    ],
    "magnitude_a": 5,
    "magnitude_b": 3,
    "normalized_a": [
      0.6,
      0.8,
      0
    ],
    "normalized_b": [
      "1/3",
      "2/3",
      "2/3"
    ],
    "projection_a_onto_b": [
      "11/9",
      "22/9",
      "22/9"
    ],
    "angle_radians": "acos(11/15) ≈ 0.7476"
  },
  "working": "dot=3×1+4×2+0×2=3+8+0=11. cross=(4×2−0×2, 0×1−3×2, 3×2−4×1)=(8,−6,2). |a|=sqrt(3²+4²+0²)=sqrt(25)=5. |b|=sqrt(1²+2²+2²)=sqrt(9)=3. normalize(a)=(3/5,4/5,0)=(0.6,0.8,0). normalize(b)=(1/3,2/3,2/3). |b|²=9, so projection coefficient=(a·b)/|b|²=11/9; projection=(11/9)(1,2,2)=(11/9,22/9,22/9). angle=acos(11/(5×3))=acos(11/15)≈0.7476 radians."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The zero vector causes division by zero in normalization, projection, and angle calculations. The instrument must reject such inputs or return an explicit undefined result. A second hazard is convention ambiguity for signed versus unsigned 2D angles.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/vector-operations-toolkit/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/vector-operations-toolkit/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/vector-operations-toolkit/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche2/tranche2-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`