Cross product

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here $E$ ), and is denoted by the symbol $\times$ . Given two linearly independent vectors $a$ and $b$ , the cross product, $a \times b$ (read "a cross b"), is a vector that is perpendicular to both $a$ and $b$ ,^[1] and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).

This article is about the cross product of two vectors in three-dimensional Euclidean space. For other uses, see Cross product (disambiguation).

The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or if either one has zero length, then their cross product is zero.^[2]

The cross product is anticommutative (that is, $a \times b = - b \times a$ ) and is distributive over addition, that is, $a \times (b + c) = a \times b + a \times c$ .^[1] The space $E$ together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.

Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation (or "handedness") of the space (it is why an oriented space is needed). The resultant vector is invariant of rotation of basis. Due to the dependence on handedness, the cross product is said to be a pseudovector.

In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space.

The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in $n$ dimensions, take the product of $n - 1$ vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.^[3] The cross-product in seven dimensions has undesirable properties, however (e.g. it fails to satisfy the Jacobi identity), so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time.^[4] (See § Generalizations below for other dimensions.)

In any basis, the cross-product $a\times b$ is given by the tensorial formula $E_{ijk}a^{i}b^{j}$ where $E_{ijk}$ is the covariant tensor (we note the position of the indices). That corresponds to the intrinsic formula given here.

Levi-Civita

In an orthonormal basis having the same orientation as the space, $a\times b$ is given by the pseudo-tensorial formula $\varepsilon _{ijk}a^{i}b^{j}$ where $\varepsilon _{ijk}$ is the Levi-Civita symbol (which is a pseudo-tensor). That is the formula used for everyday physics but it works only for this special choice of basis.

In any orthonormal basis, $a\times b$ is given by the pseudo-tensorial formula $(-1)^{B}\varepsilon _{ijk}a^{i}b^{j}$ where $(-1)^{B}=\pm 1$ indicates whether the basis has the same orientation as the space or not.

Alternative ways to compute[edit]

Conversion to matrix multiplication[edit]

The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:^[16] ${\begin{aligned}\mathbf {a} \times \mathbf {b} =[\mathbf {a} ]_{\times }\mathbf {b} &={\begin{bmatrix}\,0&\!-a_{3}&\,\,a_{2}\\\,\,a_{3}&0&\!-a_{1}\\-a_{2}&\,\,a_{1}&\,0\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\end{bmatrix}}\\\mathbf {a} \times \mathbf {b} ={[\mathbf {b} ]_{\times }}^{\mathrm {\!\!T} }\mathbf {a} &={\begin{bmatrix}\,0&\,\,b_{3}&\!-b_{2}\\-b_{3}&0&\,\,b_{1}\\\,\,b_{2}&\!-b_{1}&\,0\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}},\end{aligned}}$ where superscript $T$ refers to the transpose operation, and [a]_× is defined by: $[\mathbf {a} ]_{\times }{\stackrel {\rm {def}}{=}}{\begin{bmatrix}\,\,0&\!-a_{3}&\,\,\,a_{2}\\\,\,\,a_{3}&0&\!-a_{1}\\\!-a_{2}&\,\,a_{1}&\,\,0\end{bmatrix}}.$

The columns [a]_×,i of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with unit vectors. That is, $[\mathbf {a} ]_{\times ,i}=\mathbf {a} \times \mathbf {{\hat {e}}_{i}} ,\;i\in \{1,2,3\}$ or $[\mathbf {a} ]_{\times }=\sum _{i=1}^{3}\left(\mathbf {a} \times \mathbf {{\hat {e}}_{i}} \right)\otimes \mathbf {{\hat {e}}_{i}} ,$ where $\otimes$ is the outer product operator.

Also, if a is itself expressed as a cross product: $\mathbf {a} =\mathbf {c} \times \mathbf {d}$ then $[\mathbf {a} ]_{\times }=\mathbf {d} \mathbf {c} ^{\mathrm {T} }-\mathbf {c} \mathbf {d} ^{\mathrm {T} }.$

polar vector × polar vector = axial vector

axial vector × axial vector = axial vector

polar vector × axial vector = polar vector

axial vector × polar vector = polar vector

a $(1,n-1)$ -tensor, which takes as input $n-1$ vectors, and gives as output 1 vector – an $(n-1)$ -ary vector-valued product, or

a $(n-2,2)$ -tensor, which takes as input 2 vectors and gives as output of rank n − 2 – a binary product with rank n − 2 tensor values. One can also define $(k,n-k)$ -tensors for other k.

skew-symmetric tensor

History[edit]

In 1773, Joseph-Louis Lagrange used the component form of both the dot and cross products in order to study the tetrahedron in three dimensions.^[24]^{[note 3]}

In 1843, William Rowan Hamilton introduced the quaternion product, and with it the terms vector and scalar. Given two quaternions [0, u] and [0, v], where u and v are vectors in R³, their quaternion product can be summarized as [−u ⋅ v, u × v]. James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education.

In 1844, Hermann Grassmann published a geometric algebra not tied to dimension two or three. Grassmann developed several products, including a cross product represented then by $[uv]$ .^[25] (See also: exterior algebra.)

In 1853, Augustin-Louis Cauchy, a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product.^[26]^[27]

In 1878, William Kingdon Clifford published Elements of Dynamic, in which the term vector product is attested. In the book, this product of two vectors is defined to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane.^[28] (See also: Clifford algebra.)

In 1881 lecture notes, Gibbs represented the cross product by $u\times v$ and called it the skew product.^[29]^[30] In 1901, Gibb's student Edwin Bidwell Wilson edited and extended these lecture notes into the textbook Vector Analysis. Wilson kept the term skew product, but observed that the alternative terms cross product^{[note 4]} and vector product were more frequent.^[31]

In 1908, Cesare Burali-Forti and Roberto Marcolongo introduced the vector product notation $u \land v$ .^[25] This is used in France and other areas until this day, as the symbol $\times$ is already used to denote multiplication and the cartesian product.

– a product of two sets

Cartesian product

Geometric algebra: Rotating systems

– products involving more than three vectors

Multiple cross products

Multiplication of vectors

Quadruple product

(the symbol)

×

(1929). A History Of Mathematical Notations Volume II. Open Court Publishing. p. 134. ISBN 978-0-486-67766-8.

Cajori, Florian

Crowe, Michael J. (1994). . Dover. ISBN 0-486-67910-1.

A History of Vector Analysis

(1948) Vectorial Mechanics, Chapter 2: Vector Product, pp 11 –31, London: Methuen Publishing.

E. A. Milne

Wilson, Edwin Bidwell (1901). . Yale University Press.

Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs

T. Levi-Civita; U. Amaldi (1949). Lezioni di meccanica razionale (in Italian). Bologna: Zanichelli editore.

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Cross product"

A quick geometrical derivation and interpretation of cross products

created at Syracuse University – (requires java)

An interactive tutorial

W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).

Mathcentre (UK), 2009