Cross products #

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

Main definitions #

crossProduct is the cross product of pairs of vectors in $R^3$.

Main results #

Notation #

The locale Matrix gives the following notation:

×₃ for the cross product

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.

Instances For

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def Matrix.«term_×₃_» :

Lean.TrailingParserDescr

Instances For

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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]

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@[simp]

theorem cross_anticomm {R : Type u_1} [CommRing R] (v : Fin 3 → R) (w : Fin 3 → R) :

-↑(↑crossProduct v) w = ↑(↑crossProduct w) v

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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.

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@[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

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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 : 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.

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@[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.

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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.

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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.

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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.

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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.

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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.

Instances For

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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)

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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.