Vector Operations Toolkit

Inspectable records

Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Area

Mathematics

Tranche

tranche-2

Evidence ID

t2-045

Ledger rows

1

No ledger note is attached.

Review findings

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.

Formula and scope findings

• 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.

Unit assumptions

• 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.

Boundary hardening

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.

Errata and corrections

No applied erratum is attached to this unique-name record.

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "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."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

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

• VERIFIED_LEDGER.json · VERIFIED_CLEAN
• evidence-tranche2/tranche2-results.json
• BOUNDARY_HAZARD_CATALOG.json