Chapter 27
Vectors and Dot/Cross Product Relations

Flow Algebra

flow "Vectors and Dot/Cross Product Relations"

# ---------- CONCEPT ----------
# Vectors describe directed magnitude — the fundamental carriers of spatial coherence.
# The dot product measures alignment (scalar projection), while the cross product measures rotation (vector perpendicular to both inputs).
# Together they form the algebra of directional relation — quantifying how flows align, oppose, or rotate within space.

# ---------- STATES ----------
state VectorSystem(V):
  Vector:
    description: "v = (v₁, v₂, v₃) — ordered triple representing magnitude and direction."
  Magnitude:
    description: "‖v‖ = √(v·v) — scalar length of the vector."
  DotProduct:
    description: "u·v = Σ uᵢvᵢ = ‖u‖‖v‖cosθ — scalar measure of alignment."
  CrossProduct:
    description: "u×v — vector orthogonal to both u and v, magnitude ‖u‖‖v‖sinθ, orientation by right-hand rule."
  ScalarProjection:
    description: "proj_v(u) = (u·v)/‖v‖ — signed length of u along v."
  VectorProjection:
    description: "proj⃗_v(u) = (u·v)/(‖v‖²) v — component of u in direction v."
  TripleProduct:
    description: "(u×v)·w — scalar volume of parallelepiped spanned by u,v,w (signed)."

# ---------- PARAMETERS ----------
params:
  dimension: 3
  field: ℝ
  orthogonality: "u·v = 0"
  orientation: "Right-hand rule for cross product"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Dot product symmetric: u·v = v·u."
  - "Cross product antisymmetric: u×v = −(v×u)."
  - "Orthogonality: u·v = 0 ⇒ perpendicular vectors."
  - "Magnitude invariant under rotation: ‖R v‖ = ‖v‖ for R orthogonal."
  - "Distributivity: u·(v+w) = u·v + u·w; u×(v+w) = u×v + u×w."
  - "Scalar triple identity: (u×v)·w = (v×w)·u = (w×u)·v."
  - "Vector triple product: u×(v×w) = (u·w)v − (u·v)w."
  - "Mixed product absolute value gives volume: |(u×v)·w| = volume of parallelepiped."

# ---------- TRANSFORMS ----------
transforms:
  ComputeDot(u,v):
    rule: "u·v = u₁v₁ + u₂v₂ + u₃v₃."
  ComputeCross(u,v):
    rule: "u×v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁)."
  Magnitude(v):
    rule: "‖v‖ = √(v₁² + v₂² + v₃²)."
  ComputeAngle(u,v):
    rule: "cosθ = (u·v)/(‖u‖‖v‖)."
  Projection(u,v):
    rule: "proj⃗_v(u) = ((u·v)/(‖v‖²)) v."
  OrthogonalComponent(u,v):
    rule: "u⊥ = u − proj⃗_v(u)."
  ScalarTriple(u,v,w):
    rule: "(u×v)·w."
  VectorTriple(u,v,w):
    rule: "u×(v×w) = (u·w)v − (u·v)w."
  CheckParallel(u,v):
    condition: "u×v = 0 ⇒ vectors parallel (same or opposite direction)."
  CheckCoplanar(u,v,w):
    condition: "(u×v)·w = 0 ⇒ vectors coplanar."

# ---------- MONITORS ----------
monitors:
  OrthogonalityTest:
    measure: "u·v"
    output: "Zero ⇒ perpendicular."
  ParallelismTest:
    measure: "‖u×v‖"
    output: "Zero ⇒ parallel or antiparallel."
  AngleMonitor:
    measure: "θ = arccos((u·v)/(‖u‖‖v‖))"
    meaning: "Checks angular coherence between vectors."
  CrossMagnitudeCheck:
    measure: "‖u×v‖ − ‖u‖‖v‖|sinθ|"
    meaning: "Zero ⇒ correct cross magnitude."
  TripleProductSign:
    meaning: "Sign((u×v)·w) gives orientation (right vs left handed)."
  RotationConsistency:
    meaning: "Ensure right-hand rule maintained under transformation."

# ---------- EQUATIONS ----------
equations:
  - "u·v = u₁v₁ + u₂v₂ + u₃v₃ = ‖u‖‖v‖\\cosθ"
  - "u×v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁)"
  - "‖u×v‖ = ‖u‖‖v‖|\\sinθ|"
  - "proj_v(u) = \\frac{u·v}{‖v‖²}v"
  - "u×(v×w) = (u·w)v − (u·v)w"
  - "(u×v)·w = \\det[u,v,w]"
  - "‖u×v‖ = \\text{area of parallelogram formed by }u,v"
  - "|(u×v)·w| = \\text{volume of parallelepiped formed by }u,v,w"

# ---------- NOTES ----------
notes:
  - "The dot product captures alignment — the scalar coherence between two directions."
  - "The cross product captures rotation — the perpendicular coherence orthogonal to both directions."
  - "Dot and cross are complementary: one measures shared alignment, the other independent rotation."
  - "The scalar triple product combines them to measure volumetric coherence — how three flows enclose space."
  - "Vector triple products express nested interactions, preserving orientation and directional balance."
  - "In Flow Algebra, dot and cross represent the dual aspects of vector coherence: projection (alignment) and rotation (orthogonal generation) — the foundation of geometric interaction."