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

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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.

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      def Matrix.«term_×₃_» :
      Lean.TrailingParserDescr

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

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          theorem cross_apply {R : Type u_1} [CommRing R] (a 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]
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          @[simp]
          theorem cross_anticomm {R : Type u_1} [CommRing R] (v w : Fin 3 → R) :
          -(crossProduct v) w = (crossProduct w) v
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          theorem neg_cross {R : Type u_1} [CommRing R] (v w : Fin 3 → R) :
          -(crossProduct v) w = (crossProduct w) v

          Alias of cross_anticomm.

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

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

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

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

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

          Cyclic permutations preserve the triple product. See also triple_product_eq_det.

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          theorem triple_product_eq_det {R : Type u_1} [CommRing R] (u v w : Fin 3 → R) :
          u ⬝ᵥ (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.

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

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

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          theorem leibniz_cross {R : Type u_1} [CommRing R] (u v 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.

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          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.

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              theorem cross_cross {R : Type u_1} [CommRing R] (u v w : Fin 3 → R) :
              (crossProduct ((crossProduct u) v)) w = (crossProduct u) ((crossProduct v) w) - (crossProduct v) ((crossProduct u) w)
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              theorem jacobi_cross {R : Type u_1} [CommRing R] (u v 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.

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              theorem crossProduct_ne_zero_iff_linearIndependent {F : Type u_2} [Field F] {v w : Fin 3 → F} :
              (crossProduct v) w ≠ 0 ↔ LinearIndependent F ![v, w]
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              theorem cross_cross_eq_smul_sub_smul {R : Type u_1} [CommRing R] (u v w : Fin 3 → R) :
              (crossProduct ((crossProduct u) v)) w = (u ⬝ᵥ w) • v - (v ⬝ᵥ w) • u

              The scalar triple product expansion of the vector triple product.

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              theorem cross_cross_eq_smul_sub_smul' {R : Type u_1} [CommRing R] (u v w : Fin 3 → R) :
              (crossProduct u) ((crossProduct v) w) = (u ⬝ᵥ w) • v - (v ⬝ᵥ u) • w

              Alternative form of the scalar triple product expansion of the vector triple product.