Documentation

Mathlib.LinearAlgebra.CrossProduct

Cross products #

This module defines the cross product of vectors in $R^3$ for $R$ a commutative ring, as a bilinear map.

Main definitions #

Main results #

Notation #

The locale Matrix gives the following notation:

Tags #

crossproduct

source
def crossProduct {R : Type u_1} [CommRing R] :
(Fin 3 → R) →ₗ[R] (Fin 3 → R) →ₗ[R] Fin 3 → R

The cross product of two vectors in $R^3$ for $R$ a commutative ring.

Equations
  • crossProduct = LinearMap.mk₂ R (fun (a b : Fin 3 → R) => ![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2, a 0 * b 1 - a 1 * b 0]) ⋯ ⋯ ⋯ ⋯
Instances For
    source
    def Matrix.«term_×₃_» :
    Lean.TrailingParserDescr
    Equations
    Instances For
      source
      theorem cross_apply {R : Type u_1} [CommRing R] (a : Fin 3 → R) (b : Fin 3 → R) :
      (crossProduct a) b = ![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2, a 0 * b 1 - a 1 * b 0]
      source
      @[simp]
      theorem cross_anticomm {R : Type u_1} [CommRing R] (v : Fin 3 → R) (w : Fin 3 → R) :
      -(crossProduct v) w = (crossProduct w) v
      source
      theorem neg_cross {R : Type u_1} [CommRing R] (v : Fin 3 → R) (w : Fin 3 → R) :
      -(crossProduct v) w = (crossProduct w) v

      Alias of cross_anticomm.

      source
      @[simp]
      theorem cross_anticomm' {R : Type u_1} [CommRing R] (v : Fin 3 → R) (w : Fin 3 → R) :
      (crossProduct v) w + (crossProduct w) v = 0
      source
      @[simp]
      theorem cross_self {R : Type u_1} [CommRing R] (v : Fin 3 → R) :
      (crossProduct v) v = 0
      source
      @[simp]
      theorem dot_self_cross {R : Type u_1} [CommRing R] (v : Fin 3 → R) (w : Fin 3 → R) :
      Matrix.dotProduct v ((crossProduct v) w) = 0

      The cross product of two vectors is perpendicular to the first vector.

      source
      @[simp]
      theorem dot_cross_self {R : Type u_1} [CommRing R] (v : Fin 3 → R) (w : Fin 3 → R) :
      Matrix.dotProduct w ((crossProduct v) w) = 0

      The cross product of two vectors is perpendicular to the second vector.

      source
      theorem triple_product_permutation {R : Type u_1} [CommRing R] (u : Fin 3 → R) (v : Fin 3 → R) (w : Fin 3 → R) :
      Matrix.dotProduct u ((crossProduct v) w) = Matrix.dotProduct v ((crossProduct w) u)

      Cyclic permutations preserve the triple product. See also triple_product_eq_det.

      source
      theorem triple_product_eq_det {R : Type u_1} [CommRing R] (u : Fin 3 → R) (v : Fin 3 → R) (w : Fin 3 → R) :
      Matrix.dotProduct u ((crossProduct v) w) = Matrix.det ![u, v, w]

      The triple product of u, v, and w is equal to the determinant of the matrix with those vectors as its rows.

      source
      theorem cross_dot_cross {R : Type u_1} [CommRing R] (u : Fin 3 → R) (v : Fin 3 → R) (w : Fin 3 → R) (x : Fin 3 → R) :
      Matrix.dotProduct ((crossProduct u) v) ((crossProduct w) x) = Matrix.dotProduct u w * Matrix.dotProduct v x - Matrix.dotProduct u x * Matrix.dotProduct v w

      The scalar quadruple product identity, related to the Binet-Cauchy identity.

      source
      theorem leibniz_cross {R : Type u_1} [CommRing R] (u : Fin 3 → R) (v : Fin 3 → R) (w : Fin 3 → R) :
      (crossProduct u) ((crossProduct v) w) = (crossProduct ((crossProduct u) v)) w + (crossProduct v) ((crossProduct u) w)

      The cross product satisfies the Leibniz lie property.

      source
      def Cross.lieRing {R : Type u_1} [CommRing R] :
      LieRing (Fin 3 → R)

      The three-dimensional vectors together with the operations + and ×₃ form a Lie ring. Note we do not make this an instance as a conflicting one already exists via LieRing.ofAssociativeRing.

      Equations
      Instances For
        source
        theorem cross_cross {R : Type u_1} [CommRing R] (u : Fin 3 → R) (v : Fin 3 → R) (w : Fin 3 → R) :
        (crossProduct ((crossProduct u) v)) w = (crossProduct u) ((crossProduct v) w) - (crossProduct v) ((crossProduct u) w)
        source
        theorem jacobi_cross {R : Type u_1} [CommRing R] (u : Fin 3 → R) (v : Fin 3 → R) (w : Fin 3 → R) :
        (crossProduct u) ((crossProduct v) w) + (crossProduct v) ((crossProduct w) u) + (crossProduct w) ((crossProduct u) v) = 0

        Jacobi identity: For a cross product of three vectors, their sum over the three even permutations is equal to the zero vector.

        source
        theorem crossProduct_ne_zero_iff_linearIndependent {F : Type u_2} [Field F] {v : Fin 3 → F} {w : Fin 3 → F} :
        (crossProduct v) w ≠ 0 ↔ LinearIndependent F ![v, w]