Cross product

This article is about the cross product of two vectors. For related concepts, see cross product (disambiguation).

In mathematics, the cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, two Vectors are orthogonal if they are Perpendicular, i By contrast, the dot product produces a scalar result. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In many engineering and physics problems, it is handy to be able to construct a perpendicular vector from two existing vectors, and the cross product provides a means for doing so. The cross product is also known as the vector product, or Gibbs vector product. Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist

The cross product is not defined except in three-dimensions (and the algebra defined by the cross product is not associative). In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, associativity is a property that a Binary operation can have Like the dot product, it depends on the metric of Euclidean space. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Unlike the dot product, it also depends on the choice of orientation or "handedness". In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which Certain features of the cross product can be generalized to other situations. For arbitrary choices of orientation, the cross product must be regarded not as a vector, but as a pseudovector. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an For arbitrary choices of metric, and in arbitrary dimensions, the cross product can be generalized by the exterior product of vectors, defining a two-form instead of a vector. In Linear algebra, a two-form is another term for a Bilinear form, typically used in informal discussions or sometimes to indicate that the bilinear form is

Illustration of the cross-product in respect to a right-handed coordinate system.

Definition

Finding the direction of the cross product by the right-hand rule. For the related yet different principle relating to electromagnetic coils see Right hand grip rule.

The cross product of two vectors a and b is denoted by a × b. In physics, sometimes the notation a ∧ b is used^[1] (mathematicians do not use this notation, to avoid confusion with the exterior product). Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

In a three-dimensional Euclidean space, with a usual right-handed coordinate system, a × b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent For the related yet different principle relating to electromagnetic coils see Right hand grip rule. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides

The cross product is given by the formula

$\mathbf{a} \times \mathbf{b} = a b \sin \theta \ \mathbf{\hat{n}}$

where θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), a and b are the magnitudes of vectors a and b, and $\mathbf{\hat{n}}$ is a unit vector perpendicular to the plane containing a and b. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent If the vectors a and b are collinear (i. e. , 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.

The direction of the vector $\mathbf{\hat{n}}$ 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 $\mathbf{\hat{n}}$ is coming out of the thumb (see the picture on the right). Using this rule implies that the cross-product is anti-commutative, i. In mathematics anticommutativity refers to the property of an operation being anticommutative, i e. , b × a = - (a × b). 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.

Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector $\mathbf{\hat{n}}$ is given by the left-hand rule and points in the opposite direction. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

This, however, creates a problem because transforming from one arbitrary reference system to another (e. g. , a mirror image transformation from a right-handed to a left-handed coordinate system), should not change the direction of $\mathbf{\hat{n}}$ . The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an See cross product and handedness for more detail. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

Computing the cross product

Coordinate notation

The unit vectors i, j, and k from the given orthogonal coordinate system satisfy the following equalities:

i × j = k j × k = i k × i = j. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length

With these rules, the coordinates of the cross product of two vectors can be computed easily, without the need to determine any angles: Let

a = a₁i + a₂j + a₃k = (a₁, a₂, a₃)

and

b = b₁i + b₂j + b₃k = (b₁, b₂, b₃)

Then

a × b = (a₂b₃ − a₃b₂) i + (a₃b₁ − a₁b₃) j + (a₁b₂ − a₂b₁) k = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

Matrix notation

The definition of the cross product can also be represented by the determinant of a matrix:

$\mathbf{a}\times\mathbf{b}=\det \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\a_1 & a_2 & a_3 \\b_1 & b_2 & b_3 \\\end{bmatrix}.$

This determinant can be computed using Sarrus' rule. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the Determinant of a 3x3 Matrix. Consider the table

$\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k} & \mathbf{i} & \mathbf{j} & \mathbf{k} \\a_1 & a_2 & a_3 & a_1 & a_2 & a_3 \\b_1 & b_2 & b_3 & b_1 & b_2 & b_3 \end{matrix}$

From the first three elements on the first row draw three diagonals to the right (e. g. the first diagonal would contain i, a₂, and b₃), and from the last three elements on the first row draw three diagonals to the left (e. g. the first diagonal would contain i, a₃, and b₂). Then multiply the elements on each of these six diagonals, and negate the last three products. The cross product would be defined by the sum of these products:

$\mathbf{i}a_2b_3 + \mathbf{j}a_3b_1 + \mathbf{k}a_1b_2 - \mathbf{i}a_3b_2 - \mathbf{j}a_1b_3 - \mathbf{k}a_2b_1.$

Examples

Example 1

Consider two vectors, a = (1,2,3) and b = (4,5,6). The cross product a × b is

a × b = (1,2,3) × (4,5,6) = ((2×6 - 3×5), (3×4 - 1×6), (1×5 - 2×4)) = (-3,6,-3).

Example 2

Consider two vectors, a = (3,0,0) and b = (0,2,0). The cross product a × b is

a × b = (3,0,0) × (0,2,0) = ((0×0 - 0×2), (0×0 - 3×0), (3×2 - 0×0)) = (0,0,6).

This example has the following interpretations:

The area of the parallelogram (a rectangle in this case) is 2 × 3 = 6.
The cross product of any two vectors in the xy plane will be parallel to the z axis.
Since the z-component of the result is positive, the non-obtuse angle from a to b is counterclockwise (when observed from a point on the +z semiaxis, and when the coordinate system is right-handed).

Properties

Geometric meaning

See also: Triple_product

Figure 1: The area of a parallelogram as a cross product. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion.

Figure 2: The volume of a parallelepiped using dot and cross-products; dashed lines show the projections of c onto a × b and of a onto b × c, a first step in finding dot-products.

The magnitude of the cross product can be interpreted as the unsigned area of the parallelogram having a and b as sides (see Figure 1):

$| \mathbf{a} \times \mathbf{b}| = | \mathbf{a} | | \mathbf{b}| \sin \theta. \,\!$

Indeed, one can also compute the volume V of a parallelepiped having a, b and c as sides by using a combination of a cross product and a dot product, called scalar triple product (see Figure 2):

$V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|.$

Figure 2 demonstrates that this volume can be found in two ways, showing geometrically that the identity holds that a "dot" and a "cross" can be interchanged without changing the result. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. That is:

$V =\mathbf{a \times b \cdot c} = \mathbf{a \cdot b \times c} \ .$

Algebraic properties

The cross product is anticommutative,

a × b = −b × a,

distributive over addition,

a × (b + c) = (a × b) + (a × c),

and compatible with scalar multiplication so that

(r a) × b = a × (r b) = r (a × b). In mathematics anticommutativity refers to the property of an operation being anticommutative, i In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

It is not associative, but satisfies the Jacobi identity:

a × (b × c) + b × (c × a) + c × (a × b) = 0. In Mathematics, associativity is a property that a Binary operation can have In Mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation

It does not obey the cancellation law:

If a × b = a × c and a ≠ 0 then we can write:

(a × b) − (a × c) = 0 and, by the distributive law above:

a × (b − c) = 0

Now, if a is parallel to (b − c), then even if a ≠ 0 it is possible that (b − c) ≠ 0 and therefore that b ≠ c. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible.

However, if both a · b = a · c and a × b = a × c, then we can conclude that b = c. Indeed,

a . (b - c) = 0, and

a × (b - c) = 0

so that b - c is both parallel and perpendicular to the non-zero vector a. This is only possible if b - c = 0.

The distributivity, linearity and Jacobi identity show that R³ together with vector addition and cross product forms a Lie algebra. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In fact, the Lie algebra is that of the orthogonal group in 3 dimensions, SO(3). In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n This article is about rotations in three-dimensional Euclidean space

Further, two non-zero vectors a and b are parallel iff a × b = 0. ↔

It follows from the geometrical definition above that the cross product is invariant under rotations about the axis defined by a×b. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

Triple product expansion

Main article: Triple product

The triple product expansion, also known as Lagrange's formula, is a formula relating the cross product of three vectors (called the vector triple product) with the dot product:

a × (b × c) = b(a · c) − c(a · b). This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion.

The mnemonic “BAC minus CAB” is used to remember the order of the vectors in the right hand member. A mnemonic device (nəˈmɒnɪk is a Memory aid Commonly met mnemonics are often verbal something such as a very short poem or a special word used to help a person remember This formula is used in physics to simplify vector calculations. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A special case, regarding gradients and useful in vector calculus, is given below. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner

$\begin{align} \nabla \times (\nabla \times \mathbf{f}) & {}= \nabla (\nabla \cdot \mathbf{f} ) - (\nabla \cdot \nabla) \mathbf{f} \\& {}= \mbox{grad }(\mbox{div } \mathbf{f} ) - \mbox{laplacian } \mathbf{f}.\end{align}$

This is a special case of the more general Laplace-de Rham operator $Δ = d δ + δ d$ . In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after

The following identity also relates the cross product and the dot product:

$|\mathbf{a} \times \mathbf{b}|^2 + |\mathbf{a} \cdot \mathbf{b}|^2 = |\mathbf{a}|^2 |\mathbf{b}|^2.$

This is a special case of the multiplicativity $|\mathbf{vw}| = |\mathbf{v}| |\mathbf{w}|$ of the norm in the quaternion algebra, and a restriction to $\mathbb{R}^3$ of Lagrange's identity. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Algebra, Lagrange's identity is the identity \biggl( \sum_{k=1}^n a_k^2\biggr \biggl(\sum_{k=1}^n b_k^2\biggr - \biggl(\sum_{k=1}^n a_k b_k\biggr^2

Alternative ways to compute the cross product

Quaternions

Further information: quaternions and spatial rotation

The cross product can also be described in terms of quaternions, and this is why the letters i, j, k are a convention for the standard basis on $\mathbf{R}^3$ : it is being thought of as the imaginary quaternions. Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

Notice for instance that the above given cross product relations among i, j, and k agree with the multiplicative relations among the quaternions i, j, and k. In general, if we represent a vector [a₁, a₂, a₃] as the quaternion a₁i + a₂j + a₃k, we obtain the cross product of two vectors 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. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

Conversion to matrix multiplication

A cross product between two vectors (which can only be defined in three-dimensional space) can be rewritten in terms of pure matrix multiplication as the product of a skew-symmetric matrix and a vector, as follows:

$\mathbf{a} \times \mathbf{b} = [\mathbf{a}]_{\times} \mathbf{b} = \begin{bmatrix}\,0&\!-a_3&\,\,a_2\\ \,\,a_3&0&\!-a_1\\-a_2&\,\,a_1&\,0\end{bmatrix}\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$

$\mathbf{b} \times \mathbf{a} = [\mathbf{a}]^T_{\times} \mathbf{b} = \begin{bmatrix}\,0&\,\,a_3&\!-a_2\\ -a_3&0&\,\,a_1\\\,\,a_2&\!-a_1&\,0\end{bmatrix}\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$

where

$[\mathbf{a}]_{\times} \stackrel{\rm def}{=} \begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix}.$

Also, if $\mathbf{a}$ is itself a cross product:

$\mathbf{a} = \mathbf{c} \times \mathbf{d}$

then

$[\mathbf{a}]_{\times} = (\mathbf{c}\mathbf{d}^T)^T - \mathbf{c}\mathbf{d}^T.$

This notation provides another way of generalizing cross product to the higher dimensions by substituting pseudovectors (such as angular velocity or magnetic field) with such skew-symmetric matrices. In Linear algebra, a skew-symmetric (or antisymmetric) matrix is a Square matrix A whose Transpose is also its negative In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an Do not confuse with Angular frequency The unit for angular velocity is rad/s In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges It is clear that such physical quantities will have n(n-1)/2 independent components in n dimensions, which coincides with number of dimensions for three-dimensional space, and this is why vectors can be used (and most often are used) to represent such quantities.

This notation is also often much easier to work with, for example, in epipolar geometry. Epipolar geometry refers to the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions there are a number of geometric relations between the

From the general properties of the cross product follows immediately that

$[\mathbf{a}]_{\times} \, \mathbf{a} = \mathbf{0}$ and $\mathbf{a}^{T} \, [\mathbf{a}]_{\times} = \mathbf{0}$

and from fact that $[\mathbf{a}]_{\times}$ is skew-symmetric it follows that

$\mathbf{b}^{T} \, [\mathbf{a}]_{\times} \, \mathbf{b} = 0.$

The above-mentioned triple product expansion (bac-cab rule) can be easily proven using this notation.

The above definition of $[\mathbf{a}]_{\times}$ means that there is a one-to-one mapping between the set of 3×3 skew-symmetric matrices, also denoted SO(3), and the operation of taking the cross product with some vector $\mathbf{a}$ . This article is about rotations in three-dimensional Euclidean space

Index notation

The cross product can alternatively be defined in terms of the Levi-Civita tensor $\varepsilon_{ijk}$

$\mathbf{a \times b} = \mathbf{c}\Leftrightarrow\ c_i = \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk} a_j b_k$

where the indices $i, j, k$ correspond, as in the previous section, to orthogonal vector components. The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor

Mnemonic

The word xyzzy can be used to remember the definition of the cross product. Xyzzy is a magic word from the Colossal Cave Adventure computer game

$\mathbf{a} = \mathbf{b} \times \mathbf{c}$

where:

$\mathbf{a} = \begin{bmatrix}a_x\\a_y\\a_z\end{bmatrix}, \mathbf{b} = \begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix}, \mathbf{c} = \begin{bmatrix}c_x\\c_y\\c_z\end{bmatrix}$

then:

$a_x = b_y c_z - b_z c_y \,$

$a_y = b_z c_x - b_x c_z \,$

$a_z = b_x c_y - b_y c_x \,$

Notice that the second and third equations can be obtained from the first by simply vertically rotating the subscripts, x → y → z → x. The problem, of course, is how to remember the first equation, and two options are available for this purpose: either you remember the relevant two diagonals of Sarrus's scheme (those containing i), or you remember the xyzzy sequence. Xyzzy is a magic word from the Colossal Cave Adventure computer game

Since the first diagonal in Sarrus's scheme is just the main diagonal of the above-mentioned $3 \times 3$ matrix, the first three letters of the word xyzzy can be very easily remembered. In Linear algebra, the main diagonal (sometimes leading diagonal or primary diagonal) of a matrix A is the collection of cells A_{ij} In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which Xyzzy is a magic word from the Colossal Cave Adventure computer game

Applications

Computational geometry

The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data

In computational geometry of the plane, the cross product is used to determine the sign of the acute angle defined by three points $p 1 = (x 1, y 1)$ , $p 2 = (x 2, y 2)$ and $p 3 = (x 3, y 3)$ . Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called It corresponds to the direction of the cross product of the two coplanar vectors defined by the pairs of points $p 1, p 2$ and $p 1, p 3$ , i. e. , by the sign of the expression $P = (x 2 - x 1)(y 3 - y 1) - (y 2 - y 1)(x 3 - x 1)$ . In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a negative angle of rotation around $p 2$ from $p 1$ to $p 3$ , otherwise a positive angle. From another point of view, the sign of $P$ tells whether $p 3$ lies to the left or to the right of line $p 1, p 2$ .

Other

The cross product occurs in the formula for the vector operator curl. A vector operator is a type of Differential operator used in Vector calculus. cURL is a Command line tool for transferring files with URL syntax. It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation The definitions of torque and angular momentum also involve the cross product. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi-view geometry, in particular when deriving matching constraints.

Cross product as an exterior product

The cross product in relation to the exterior product. In red are the unit normal vector, and the "parallel" unit bivector.

The cross product can be viewed in terms of the exterior product. This view allows for a natural geometric interpretation of the cross product. In exterior calculus the exterior product (or wedge product) of two vectors is a bivector. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p 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 a∧b as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge dual of the bivector a∧b, identifying 2-vectors with vectors:

$a \times b = * (a \wedge b) \,.$

This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p Only in three dimensions is the result an oriented line element – a vector – whereas, for example, in 4 dimensions the Hodge dual of a bivector is two-dimensional – another oriented plane element. So, in three dimensions only is the cross product of a and b the vector dual to the bivector a∧b: it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as a∧b has relative to the unit bivector; precisely the properties described above.

Cross product and handedness

When measurable quantities involve cross products, the handedness of the coordinate systems used cannot be arbitrary. However, when physics laws are written as equations, it should be possible to make an arbitrary choice of the coordinate system (including handedness). To avoid problems, one should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two vectors, one must take into account that when the handedness of the coordinate system is not fixed a priori, the result is not a (true) vector but a pseudovector. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an Therefore, for consistency, the other side must also be a pseudovector.

More generally, the result of a cross product may be either a vector or a pseudovector, depending on the type of its operands (vectors or pseudovectors). Namely, vectors and pseudovectors are interrelated in the following ways under application of the cross product:

vector × vector = pseudovector
vector × pseudovector = vector
pseudovector × pseudovector = pseudovector

Because the cross product may also be a (true) vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a (true) vector and the other one is a pseudovector (e. g. , the cross product of two vectors). For instance, a vector triple product involving three (true) vectors is a (true) vector. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion.

A handedness-free approach is possible using exterior algebra.

Higher dimensions

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

In the context of multilinear algebra, it is possible to define a generalized cross product in terms of parity such that the generalized cross product between two vectors of dimension n is a skew-symmetric tensor of rank n−2. In Mathematics, multilinear algebra extends the methods of Linear algebra.

Using octonions

Main article: seven-dimensional cross product

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. In Mathematics, the seven-dimensional cross product is a Binary operation on vectors in a seven-dimensional Euclidean space. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real The nonexistence of such cross products of two vectors in other dimensions is related to the result that the only normed division algebras are the ones with dimension 1, 2, 4, and 8. In Mathematics, a normed division algebra A is a Division algebra over the real or complex numbers which is also a Normed vector

Wedge product

Main article: Exterior algebra

In general dimension, there is no direct analogue of the binary cross product. There is however the wedge product, which has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector. In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p As mentioned above, the cross product can be interpreted as the wedge product in three dimensions after using Hodge duality to identify 2-vectors with vectors.

One can also construct an n-ary analogue of the cross product in Rⁿ⁺¹ given by

$\bigwedge(\mathbf{v}_1,\cdots,\mathbf{v}_n)=\begin{vmatrix} v_1{}^1 &\cdots &v_1{}^{n+1}\\\vdots &\ddots &\vdots\\v_n{}^1 & \cdots &v_n{}^{n+1}\\\mathbf{e}_1 &\cdots &\mathbf{e}_{n+1}\end{vmatrix}.$

This formula is identical in structure to the determinant formula for the normal cross product in R³ 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 (v₁,. . . ,v_n,Λ(v₁,. . . ,v_n)) have a positive orientation with respect to (e₁,. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which . . ,e_n+1). If n is even, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is odd, however, the distinction must be kept. This n-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.

The wedge product and dot product can be combined to form the Clifford product. In Mathematics, Clifford algebras are a type of Associative algebra.

History

In 1773, Joseph Louis Lagrange introduced the component form of both the dot and cross products in order to study the tetrahedron in three dimensions. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. ^[2] In 1843 the Irish mathematical physicist Sir William Rowan Hamilton introduced the quaternion product, and with it the terms "vector" and "scalar". Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician 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. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

However, Oliver Heaviside in England and Josiah Willard Gibbs in Connecticut felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist Connecticut ( is a state located in the New England region of the northeastern United States of America. Thus, about forty years after the quaternion product, the dot product and cross product were introduced — to heated opposition. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R Pivotal to (eventual) acceptance was the efficiency of the new approach, allowing Heaviside to reduce the equations of electromagnetism from Maxwell's original 20 to the four commonly seen today.

Largely independent of this development, and largely unappreciated at the time, Hermann Grassmann created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a William Kingdon Clifford combined the algebras of Hamilton and Grassmann to produce Clifford algebra, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product. William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and In Mathematics, Clifford algebras are a type of Associative algebra.

The cross notation, which began with Gibbs, inspired the name "cross product". Originally appearing in privately published notes for his students in 1881 as Elements of Vector Analysis, Gibbs’s notation — and the name — later reached a wider audience through Vector Analysis (Gibbs/Wilson), a textbook by a former student. Vector Analysis is a book on Vector calculus first published in 1901 by Edwin Bidwell Wilson. Edwin Bidwell Wilson rearranged material from Gibbs's lectures, together with material from publications by Heaviside, Föpps, and Hamilton. Edwin Bidwell Wilson ( April 25 1879 – December 28 1964) was a Mathematician and Polymath. He divided vector analysis into three parts:

"First, that which concerns addition and the scalar and vector products of vectors. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function. "

Two main kinds of vector multiplications were defined, and they were called as follows:

The direct, scalar, or dot product of two vectors
The skew, vector, or cross product of two vectors

Several kinds of triple products and products of more than three vectors were also examined. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. The above mentioned triple product expansion was also included.

Notes

^ Jeffreys, H and Jeffreys, BS (1999). The multiplication sign is the symbol × ( multiplication sign is the preferred Unicode name for the Codepoint represented by that Glyph Methods of mathematical physics. Cambridge University Press.
^ Lagrange, JL (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires", Oeuvres vol 3.

References

Cajori, Florian (1929), A History Of Mathematical Notations Volume II, Open Court Publishing, p. Florian Cajori ( February 28 1859 in St Aignan (near Thusis Graubünden, Switzerland &mdash August 15, 1930, Berkeley The Open Court Publishing Company is a Publisher with offices in Chicago and La Salle Illinois. p. 134, ISBN 978-0-486-67766-8, <http://www.archive.org/details/historyofmathema027671mbp>
Wilson, Edwin Bidwell (1901), Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs, Yale University Press, <http://www.archive.org/details/117714283>

External links

Eric W. Weisstein, Cross Product at MathWorld. Yale University Press is a book Publisher founded in 1908. It became an official department of Yale University in 1961, but remained Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Z.K. Silagadze (2002). Multi-dimensional vector product. Journal of Physics. A35, 4949 (it is only possible in 7-D space)
Real and Complex Products of Complex Numbers
Vector Product Calculator Online application to calculate the vector product of 3 element vectors
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).

Syracuse University (SU is a private research university located in Syracuse, New York.

Dictionary

-noun

(vector algebra) A vector product.

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