thumb|right|150px|The cross product with respect to a right-handed coordinate system

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 <math>E</math>), and is denoted by the symbol <math>\times</math>. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , 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).

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.

The cross product is anticommutative (that is, ) and is distributive over addition, that is, . The cross-product in seven dimensions has undesirable properties (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. (See below for other dimensions.)

Definition

thumb|Finding the direction of the cross product by the [[right-hand rule ]]

The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by . In physics and applied mathematics, the wedge notation is often used (in conjunction with the name vector product), although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to dimensions.

The cross product is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule

<math display="block">\mathbf{a} \times \mathbf{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin(\theta) \, \mathbf{n},</math>

where

If the vectors a and b are parallel (that is, the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

Direction

left|thumb|The cross product (vertical, in purple) changes as the angle between the vectors a (blue) and b (red) changes. The cross product is always orthogonal to both vectors, and has magnitude zero when the vectors are parallel and maximum magnitude ‖a‖‖b‖ when they are orthogonal.

The direction of the vector n depends on the chosen orientation of the space. Conventionally, it is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is anti-commutative; that is, . By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector.

As the cross product operator depends on the orientation of the space, in general the cross product of two vectors is not a "true" vector, but a pseudovector.

Names and origin

upright=1.25|thumb|right|According to [[Sarrus's rule, the determinant of a 3×3 matrix involves multiplications between matrix elements identified by crossed diagonals]]

In 1842, William Rowan Hamilton first described the algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero scalar part) is performed, it results in a quaternion with a scalar and vector part. The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors.

In 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced the notation for both the dot product and the cross product using a period () and an "×" (), respectively, to denote them.

In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford coined the alternative names scalar product and vector product for the two operations.]]

thumb|350px|The two nonequivalent triple cross products of three vectors a, b, c. In each case, two vectors define a plane, the other is out of the plane and can be split into parallel and perpendicular components to the cross product of the vectors defining the plane. These components can be found by [[vector projection and rejection. The triple product is in the plane and is rotated as shown.]]

If the cross product of two vectors is the zero vector (that is, ), then either one or both of the inputs is the zero vector, ( or ) or else they are parallel or antiparallel () so that the sine of the angle between them is zero ( or and ).

The self cross product of a vector is the zero vector:

<math display="block">\mathbf{a} \times \mathbf{a} = \mathbf{0}.</math>

The cross product is anticommutative,

<math display="block">\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}),</math>

distributive over addition,

<math display="block">\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) + (\mathbf{a} \times \mathbf{c}),</math>

and compatible with scalar multiplication so that

<math display="block">(r\,\mathbf{a}) \times \mathbf{b} = \mathbf{a} \times (r\,\mathbf{b}) = r\,(\mathbf{a} \times \mathbf{b}).</math>

It is not associative, but satisfies the Jacobi identity:

<math display="block">\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0}.</math>

Distributivity, linearity and Jacobi identity show that the R<sup>3</sup> vector space together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO(3).

The cross product does not obey the cancellation law; that is, with does not imply , but only that:

<math display="block"> \begin{align}

\mathbf{0} &= (\mathbf{a} \times \mathbf{b}) - (\mathbf{a} \times \mathbf{c})\\

&= \mathbf{a} \times (\mathbf{b} - \mathbf{c}).

\end{align}</math>

This can be the case where b and c cancel, but additionally where a and are parallel; that is, they are related by a scale factor t, leading to:

<math display="block">\mathbf{c} = \mathbf{b} + t\,\mathbf{a},</math>

for some scalar t.

If, in addition to and as above, it is the case that then

<math display="block">\begin{align}

\mathbf{a} \times (\mathbf{b} - \mathbf{c}) &= \mathbf{0} \\

\mathbf{a} \cdot (\mathbf{b} - \mathbf{c}) &= 0,

\end{align}</math>

As cannot be simultaneously parallel (for the cross product to be 0) and perpendicular (for the dot product to be 0) to a, it must be the case that b and c cancel: .

From the geometrical definition, the cross product is invariant under proper rotations about the axis defined by . In formulae:

<math display="block">(R\mathbf{a}) \times (R\mathbf{b}) = R(\mathbf{a} \times \mathbf{b}),</math> where <math>R</math> is a rotation matrix with <math>\det(R)=1</math>.

More generally, the cross product obeys the following identity under matrix transformations:

<math display="block">(M\mathbf{a}) \times (M\mathbf{b}) = (\det M) \left(M^{-1}\right)^\mathrm{T}(\mathbf{a} \times \mathbf{b}) = \operatorname{cof} M (\mathbf{a} \times \mathbf{b}) </math>

where <math>M</math> is a 3-by-3 matrix and <math>\left(M^{-1}\right)^\mathrm{T}</math> is the transpose of the inverse and <math>\operatorname{cof}</math> is the cofactor matrix. It can be readily seen how this formula reduces to the former one if <math>M</math> is a rotation matrix. If <math>M</math> is a 3-by-3 symmetric matrix applied to a generic cross product <math>\mathbf{a} \times \mathbf{b}</math>, the following relation holds true:

<math display="block">M(\mathbf{a} \times \mathbf{b}) = \operatorname{Tr}(M)(\mathbf{a} \times \mathbf{b}) - \mathbf{a} \times M\mathbf{b} + \mathbf{b} \times M\mathbf{a}</math>

The cross product of two vectors lies in the null space of the matrix with the vectors as rows:

<math display="block">\mathbf{a} \times \mathbf{b} \in NS\left(\begin{bmatrix}\mathbf{a} \\ \mathbf{b}\end{bmatrix}\right).</math>

For the sum of two cross products, the following identity holds:

<math display="block">\mathbf{a} \times \mathbf{b} + \mathbf{c} \times \mathbf{d} = (\mathbf{a} - \mathbf{c}) \times (\mathbf{b} - \mathbf{d}) + \mathbf{a} \times \mathbf{d} + \mathbf{c} \times \mathbf{b}.</math>

Differentiation

The product rule of differential calculus applies to any bilinear operation, and therefore also to the cross product:

<math display="block">\frac{d}{dt}(\mathbf{a} \times \mathbf{b}) = \frac{d\mathbf{a{dt} \times \mathbf{b} + \mathbf{a} \times \frac{d\mathbf{b{dt} ,</math>

where a and b are vectors that depend on the real variable t.

Triple product expansion

The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as

<math display="block">\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}), </math>

It is the signed volume of the parallelepiped with edges a, b and c and as such the vectors can be used in any order that's an even permutation of the above ordering. The following therefore are equal:

<math display="block">\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}), </math>

The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula

<math display="block">\begin{align}

\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) \\

(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \mathbf{b}(\mathbf{c} \cdot \mathbf{a}) - \mathbf{a} (\mathbf{b} \cdot \mathbf{c})

\end{align}</math>

The mnemonic "BAC minus CAB" is used to remember the order of the vectors in the right hand member. This formula is used in physics to simplify vector calculations. A special case, regarding gradients and useful in vector calculus, is

<math display="block">\begin{align}

\nabla \times (\nabla \times \mathbf{f}) &= \nabla (\nabla \cdot \mathbf{f} ) - (\nabla \cdot \nabla) \mathbf{f} \\

&= \nabla (\nabla \cdot \mathbf{f} ) - \nabla^2 \mathbf{f},\\

\end{align}</math>

where ∇<sup>2</sup> is the vector Laplacian operator.

Other identities relate the cross product to the scalar triple product:

<math display="block">\begin{align}

(\mathbf{a}\times \mathbf{b})\times (\mathbf{a}\times \mathbf{c}) &= (\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})) \mathbf{a} \\

(\mathbf{a}\times \mathbf{b})\cdot(\mathbf{c}\times \mathbf{d}) &= \mathbf{b}^\mathrm{T} \left( \left( \mathbf{c}^\mathrm{T} \mathbf{a}\right)I - \mathbf{c} \mathbf{a}^\mathrm{T} \right) \mathbf{d}\\ &= (\mathbf{a}\cdot \mathbf{c})(\mathbf{b}\cdot \mathbf{d})-(\mathbf{a}\cdot \mathbf{d}) (\mathbf{b}\cdot \mathbf{c})

\end{align}</math>

where I is the identity matrix.

Alternative formulation

The cross product and the dot product are related by:

<math display="block"> \left\| \mathbf{a} \times \mathbf{b} \right\|^2 = \left\| \mathbf{a}\right\|^2 \left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2 .</math>

The right-hand side is the Gram determinant of a and b, the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle θ between the two vectors, as:

<math display="block"> \mathbf{a \cdot b} = \left\| \mathbf a \right\| \left\| \mathbf b \right\| \cos \theta , </math>

the above given relationship can be rewritten as follows:

<math display="block"> \left\| \mathbf{a \times b} \right\|^2 = \left\| \mathbf{a} \right\| ^2 \left\| \mathbf{b}\right \| ^2 \left(1-\cos^2 \theta \right) .</math>

Invoking the Pythagorean trigonometric identity one obtains:

<math display="block"> \left\| \mathbf{a} \times \mathbf{b} \right\| = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \left| \sin \theta \right| ,</math>

which is the magnitude of the cross product expressed in terms of θ, equal to the area of the parallelogram defined by a and b (see definition above).

The combination of this requirement and the property that the cross product be orthogonal to its constituents a and b provides an alternative definition of the cross product.

Cross product inverse

Given two vectors and with , the equation admits solutions for if and only if is orthogonal to (that is, if ). In that case, there exists an infinite family of solutions for , which are

<math display="block"> \mathbf{b} = \frac{\mathbf{c} \times \mathbf{a{\left\| \mathbf{a} \right\|^2} + t \mathbf{a} ,</math>

where is an arbitrary constant.

This can be derived using the triple product expansion:

<math display="block">

\mathbf{c} \times \mathbf{a} = (\mathbf{a} \times \mathbf{b}) \times \mathbf{a}

= \left\| \mathbf{a} \right\|^2 \mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{a} </math>

Rearrange to solve for to give

<math display="block">

\mathbf{b} = \frac{\mathbf{c} \times \mathbf{a{\left\| \mathbf{a} \right\|^2} + \frac{\mathbf{a}\cdot \mathbf{b{\left\| \mathbf{a} \right\|^2}\mathbf{a}

</math>

The coefficient of the last term can be simplified to just the arbitrary constant to yield the result shown above.

Lagrange's identity

The relation

<math display="block">

\left\| \mathbf{a} \times \mathbf{b} \right\|^2 =

\det \begin{bmatrix}

\mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} \\

\mathbf{a} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{b}

\end{bmatrix} =

\left\| \mathbf{a} \right\| ^2 \left\| \mathbf{b} \right\| ^2 - (\mathbf{a} \cdot \mathbf{b})^2

</math>

can be compared with another relation involving the right-hand side, namely Lagrange's identity expressed as

<math display="block">

\sum_{1 \le i < j \le n} \left( a_ib_j - a_jb_i \right)^2 =

\left\| \mathbf a \right\|^2 \left\| \mathbf b \right\|^2 - ( \mathbf{a \cdot b } )^2,

</math>

where a and b may be n-dimensional vectors. This also shows that the Riemannian volume form for surfaces is exactly the surface element from vector calculus. In the case where , combining these two equations results in the expression for the magnitude of the cross product in terms of its components:

<math display="block">\begin{align}

\|\mathbf{a} \times \mathbf{b}\|^2

&= \sum_{1 \le i < j \le 3} (a_ib_j - a_jb_i)^2 \\

&= (a_1 b_2 - b_1 a_2)^2 + (a_2 b_3 - a_3 b_2)^2 + (a_3 b_1 - a_1 b_3)^2.

\end{align}</math>

The same result is found directly using the components of the cross product found from

<math display="block">\mathbf{a} \times \mathbf{b} = \det \begin{bmatrix}

\hat\mathbf{i} & \hat\mathbf{j} & \hat\mathbf{k} \\

a_1 & a_2 & a_3 \\

b_1 & b_2 & b_3 \\

\end{bmatrix}.</math>

In R<sup>3</sup>, Lagrange's equation is a special case of the multiplicativity of the norm in the quaternion algebra.

It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the Binet–Cauchy identity:

<math display="block">

(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) =

(\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}).

</math>

If and , this simplifies to the formula above.

Alternative ways to compute

Conversion to matrix multiplication

The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:

<math display="block">\begin{align}

\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{align}</math>

where the superscript refers to the transpose operation, and [a]<sub>×</sub> is defined by

<math display="block">[\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}.</math>

The columns [a]<sub>×,i</sub> of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with unit vectors. That is,

<math display="block">[\mathbf{a}]_{\times, i} = \mathbf{a} \times \mathbf{\hat{e}_i}, \; i\in \{1,2,3\} </math>

or

<math display="block">[\mathbf{a}]_{\times} = \sum_{i=1}^3\left(\mathbf{a} \times \mathbf{\hat{e}_i}\right)\otimes\mathbf{\hat{e}_i},</math>

where <math>\otimes</math> is the outer product operator.

Also, if a is itself expressed as a cross product:

<math display="block">\mathbf{a} = \mathbf{c} \times \mathbf{d}</math>

then

<math display="block">[\mathbf{a}]_{\times} = \mathbf{d}\mathbf{c}^\mathrm{T} - \mathbf{c}\mathbf{d}^\mathrm{T} .</math>

This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector. In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product, with the bivector instead of its vector dual. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors. This generalization allows a natural geometric interpretation of the cross product. In exterior algebra the exterior product of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors a and b, one can view the bivector as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge star of the bivector , mapping 2-vectors to vectors:

<math display="block">a \times b = \star (a \wedge b).</math>

This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. In a d-dimensional space, Hodge star takes a k-vector to a (d–k)-vector; thus only in d = 3 dimensions is the result an element of dimension one (3–2 = 1), i.e. a vector. For example, in d = 4 dimensions, the cross product of two vectors has dimension 4–2 = 2, giving a bivector. Thus, only in three dimensions does cross product define an algebra structure to multiply vectors.

Generalizations

There are several ways to generalize the cross product to higher dimensions.

Lie algebra

The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras, which are axiomatized as binary products satisfying the axioms of multilinearity, skew-symmetry, and the Jacobi identity. Many Lie algebras exist, and their study is a major field of mathematics, called Lie theory.

For example, the Heisenberg algebra gives another Lie algebra structure on <math>\mathbf{R}^3.</math> In the basis <math>\{x,y,z\},</math> the product is <math>[x,y]=z, [x,z]=[y,z]=0.</math>

Quaternions

The cross product can also be described in terms of quaternions.

In general, if a vector is represented as the quaternion , the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors.

Octonions

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector-valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem that the only normed division algebras are the ones with dimension 1, 2, 4, and 8.

Exterior product

In general dimension, there is no direct analogue of the binary cross product that yields specifically a vector. There is however the exterior product, which has similar properties, except that the exterior product of two vectors is now a 2-vector instead of an ordinary vector. As mentioned above, the cross product can be interpreted as the exterior product in three dimensions by using the Hodge star operator to map 2-vectors to vectors. The Hodge dual of the exterior product yields an -vector, which is a natural generalization of the cross product in any number of dimensions.

The exterior product and dot product can be combined (through summation) to form the geometric product in geometric algebra.

External product

As mentioned above, the cross product can be interpreted in three dimensions as the Hodge dual of the exterior product. In any finite n dimensions, the Hodge dual of the exterior product of vectors is a vector. So, instead of a binary operation, in arbitrary finite dimensions, the cross product is generalized as the Hodge dual of the exterior product of some given vectors. This generalization is called external product.

Commutator product

Interpreting the three-dimensional vector space of the algebra as the 2-vector (not the 1-vector) subalgebra of the three-dimensional geometric algebra, where <math>\mathbf{i} = \mathbf{e_2} \mathbf{e_3}</math>, <math>\mathbf{j} = \mathbf{e_1} \mathbf{e_3}</math>, and <math>\mathbf{k} = \mathbf{e_1} \mathbf{e_2}</math>, the cross product corresponds exactly to the commutator product in geometric algebra and both use the same symbol <math>\times</math>. The commutator product is defined for 2-vectors <math>A</math> and <math>B</math> in geometric algebra as:

<math display="block">A \times B = \tfrac{1}{2}(AB - BA),</math>

where <math>AB</math> is the geometric product.

The commutator product could be generalised to arbitrary multivectors in three dimensions, which results in a multivector consisting of only elements of grades 1 (1-vectors/true vectors) and 2 (2-vectors/pseudovectors). While the commutator product of two 1-vectors is indeed the same as the exterior product and yields a 2-vector, the commutator of a 1-vector and a 2-vector yields a true vector, corresponding instead to the left and right contractions in geometric algebra. The commutator product of two 2-vectors has no corresponding equivalent product, which is why the commutator product is defined in the first place for 2-vectors. Furthermore, the commutator triple product of three 2-vectors is the same as the vector triple product of the same three pseudovectors in vector algebra. However, the commutator triple product of three 1-vectors in geometric algebra is instead the negative of the vector triple product of the same three true vectors in vector algebra.

Generalizations to higher dimensions is provided by the same commutator product of 2-vectors in higher-dimensional geometric algebras, but the 2-vectors are no longer pseudovectors. Just as the commutator product/cross product of 2-vectors in three dimensions correspond to the simplest Lie algebra, the 2-vector subalgebras of higher dimensional geometric algebra equipped with the commutator product also correspond to the Lie algebras. Also as in three dimensions, the commutator product could be further generalised to arbitrary multivectors.

Multilinear algebra

In the context of multilinear algebra, the cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form, a (0,3)-tensor, by raising an index.

In detail, the 3-dimensional volume form defines a product <math> V \times V \times V \to \mathbf{R},</math> by taking the determinant of the matrix given by these 3 vectors. By duality, this is equivalent to a function <math> V \times V \to V^*,</math> (fixing any two inputs gives a function <math> V \to \mathbf{R}</math> by evaluating on the third input) and in the presence of an inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism <math> V \to V^*,</math> and thus this yields a map <math> V \times V \to V,</math> which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index".

Translating the above algebra into geometry, the function "volume of the parallelepiped defined by <math> (a,b,-)</math>" (where the first two vectors are fixed and the last is an input), which defines a function <math> V \to \mathbf{R}</math>, can be represented uniquely as the dot product with a vector: this vector is the cross product <math> a \times b.</math> From this perspective, the cross product is defined by the scalar triple product, <math>\mathrm{Vol}(a,b,c) = (a\times b)\cdot c.</math>

In the same way, in higher dimensions one may define generalized cross products by raising indices of the n-dimensional volume form, which is a <math> (0,n)</math>-tensor.

The most direct generalizations of the cross product are to define either:

  • a <math> (1,n-1)</math>-tensor, which takes as input <math> n-1</math> vectors, and gives as output 1 vector – an <math> (n-1)</math>-ary vector-valued product, or
  • a <math> (n-2,2)</math>-tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank – a binary product with rank tensor values. One can also define <math>(k,n-k)</math>-tensors for other k.

These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity.

The <math> (n-1)</math>-ary product can be described as follows: given <math> n-1</math> vectors <math> v_1,\dots,v_{n-1}</math> in <math>\mathbf{R}^n,</math> define their generalized cross product <math> v_n = v_1 \times \cdots \times v_{n-1}</math> as:

  • perpendicular to the hyperplane defined by the <math> v_i,</math>
  • magnitude is the volume of the parallelotope defined by the <math> v_i,</math> which can be computed as the Gram determinant of the <math> v_i,</math>
  • oriented so that <math> v_1,\dots,v_n</math> is positively oriented.

This is the unique multilinear, alternating product which evaluates to <math> e_1 \times \cdots \times e_{n-1} = e_n</math>, <math> e_2 \times \cdots \times e_n = e_1,</math> and so forth for cyclic permutations of indices.

In coordinates, one can give a formula for this <math> (n-1)</math>-ary analogue of the cross product in R<sup>n</sup> by:

<math display="block">\bigwedge_{i=0}^{n-1}\mathbf{v}_i =

\begin{vmatrix}

v_1{}^1 &\cdots &v_1{}^{n}\\

\vdots &\ddots &\vdots\\

v_{n-1}{}^1 & \cdots &v_{n-1}{}^{n}\\

\mathbf{e}_1 &\cdots &\mathbf{e}_{n}

\end{vmatrix}.

</math>

This formula is identical in structure to the determinant formula for the normal cross product in R<sup>3</sup> except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v<sub>1</sub>, ..., v<sub>n−1</sub>, Λv<sub>i</sub>) have a positive orientation with respect to (e<sub>1</sub>, ..., e<sub>n</sub>). If n is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is even, however, the distinction must be kept. This <math> (n-1)</math>-ary form enjoys many of the same properties as the vector cross product: it is alternating and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments. Moreover, the product <math>[v_1,\ldots,v_n]:=\bigwedge_{i=0}^n v_i</math> satisfies the Filippov identity,

<math display="block">

[[x_1,\ldots,x_n],y_2,\ldots,y_n]] = \sum_{i=1}^n [x_1,\ldots,x_{i-1},[x_i,y_2,\ldots,y_n],x_{i+1},\ldots,x_n],

</math>

and so it endows R<sup>n+1</sup> with a structure of n-Lie algebra (see Proposition 1 of ).

History

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.

In 1843, William Rowan Hamilton introduced the quaternion product, and with it the terms vector and scalar. Given two quaternions and , where u and v are vectors in R<sup>3</sup>, their quaternion product can be summarized as . 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 . (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.

In 1878, William Kingdon Clifford, known for a precursor to the Clifford algebra named in his honor, 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.

In lecture notes from 1881, Gibbs represented the cross product by <math>u \times v</math> and called it the skew product. 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 and vector product were more frequent.

In 1908, Cesare Burali-Forti and Roberto Marcolongo introduced the vector product notation . This is used in France and other areas until this day, as the symbol <math>\times</math> is already used to denote multiplication and the Cartesian product.

See also

  • Cartesian product – a product of two sets
  • Geometric algebra: Rotating systems
  • Multiple cross products – products involving more than three vectors
  • Multiplication of vectors
  • Quadruple product
  • × (the symbol)

Notes

References

Bibliography

  • E. A. Milne (1948) Vectorial Mechanics, Chapter 2: Vector Product, pp 11 –31, London: Methuen Publishing.
  • A quick geometrical derivation and interpretation of cross products
  • An interactive tutorial created at Syracuse University – (requires java)
  • W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).
  • The vector product, Mathcentre (UK), 2009