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Encyclopedia > Cross product

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. By contrast, the dot product produces a scalar result. 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. In mathematics, the Künneth theorem of algebraic topology describes the singular homology of the cartesian product X × Y of two topological spaces, in terms of singular homology groups Hi(X, R) and Hj(X, R). ... In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. ... In physics and engineering, a vector is a physical entity which has a magnitude which is a scalar (a physical quantity expressed as the product of a numerical value and a physical unit, not just a number). ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... Josiah Willard Gibbs (February 11, 1839 New Haven – April 28, 1903 New Haven) was one of the very first American theoretical physicists and chemists. ...


The cross product is not defined except in three-dimensions (and the algebra defined by the cross product is not associative). Like the dot product, it depends on the metric of Euclidean space. Unlike the dot product, it also depends on the choice of orientation or "handedness". 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. 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 mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

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

Contents

-from the creater, wshun 02:20, 24 Dec 2004 (UTC) File links The following pages link to this file: Vector (spatial) Cross product Categories: FAL images ... -from the creater, wshun 02:20, 24 Dec 2004 (UTC) File links The following pages link to this file: Vector (spatial) Cross product Categories: FAL images ...

Definition

Finding the direction of the cross product by the right-hand rule.
Finding the direction of the cross product by the right-hand rule.

The cross product of two vectors a and b is denoted by a × b. In a three-dimensional Euclidean space, with a usual right-handed coordinate system, it 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. Image File history File links Size of this preview: 660 × 600 pixelsFull resolution (1086 × 987 pixel, file size: 79 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Size of this preview: 660 × 600 pixelsFull resolution (1086 × 987 pixel, file size: 79 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The left-handed orientation is shown on the left, and the right-handed on the right. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Fig. ... Fig. ... The left-handed orientation is shown on the left, and the right-handed on the right. ... A parallelogram. ...


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 (non-obtuse) 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. 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 angle symbol. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... Fig. ...


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 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. Fig. ...


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. See cross product and handedness for more detail. In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... For the crossed product in algebra and functional analysis, see crossed product. ...


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),(6 × 1 - 4 × 3), (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:

  1. The area of the parallelogram (a rectangle in this case) is 2 × 3 = 6.
  2. The cross product of any two vectors in the xy plane will be parallel to the z axis.
  3. 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

The area of a parallelogram as a cross product.
The area of a parallelogram as a cross product.

The magnitude of the cross product can be interpreted as the unsigned area of the parallelogram having a and b as sides: Image File history File links Size of this preview: 685 × 599 pixelsFull resolution (1110 × 971 pixel, file size: 51 KB, MIME type: image/png) <?xml version=1. ... Image File history File links Size of this preview: 685 × 599 pixelsFull resolution (1110 × 971 pixel, file size: 51 KB, MIME type: image/png) <?xml version=1. ... Area is a physical quantity expressing the size of a part of a surface. ... A parallelogram. ...

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

Algebraic properties

The cross product is anticommutative, A mathematical operator (typically a binary operator, represented by *) is anticommutative iff it is true that x * y = &#8722;(y * x) for all x and y on the operators valid domain (e. ...

a × b = −b × a,

distributive over addition, In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

a × (b + c) = (a × b) + (a × c),

and compatible with scalar multiplication so that

(ra) × b = a × (rb) = r(a × b).

It is not associative, but satisfies the Jacobi identity: 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. ...

a × (b × c) + b × (c × a) + c × (a × b) = 0.

It does not obey the cancellation law: In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. ...

If a × b = a × c and a0 then we can write:
(a × b) − (a × c) = 0 and, by the distributive law above:
a × (bc) = 0
Now, if a is parallel to (bc), then even if a0 it is possible that (bc) ≠ 0 and therefore that bc.

However, if both a · b = a · c and a × b = a × c, then we can conclude that b = c. This is because if (bc) ≠ 0, then it obviously cannot be both parallel and perpendicular to another nonzero vector a.


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. ...


Further, two non-zero vectors a and b are parallel iff a × b = 0. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...


Lagrange's formula (triple product expansion)

Main article: Lagrange's formula

This is a very useful identity involving the cross-product. It is written as This is a well-known and useful formula, a × (b × c) = b(a · c) − c(a · b), which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. ...

a × (b × c) = b(a · c) − c(a · b),

which is easier to remember as “BAC minus CAB”, keeping in mind which vectors are dotted together. This formula is very useful in simplifying vector calculations in physics. A special case, regarding gradients and useful in vector calculus, is given below. This is a discussion of a present category of science. ... For other uses, see Gradient (disambiguation). ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

 begin{matrix} 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{matrix}

This is a special case of the more general Laplace-de Rham operator Δ = dδ + δd. In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...


Another useful identity of Lagrange is

 |a times b|^2 + |a cdot b|^2 = |a|^2 |b|^2.

This is a special case of the multiplicativity | vw | = | v | | w | of the norm in the quaternion algebra. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...


Ways to compute a cross product

Coordinate notation

The unit vectors i, j, and k from the given orthogonal coordinate system satisfy the following equalities: In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

i × j = k           j × k = i           k × i = j.

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 coordinate notation can also be written formally as the determinant of a matrix: In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... For the square matrix section, see square matrix. ...

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

The determinant of three vectors can be recovered as

det (a, b, c) = a · (b × c).

Intuitively, the cross product can be described by Sarrus's scheme. Consider the table Pierre Frédéric Sarrus (10 March 1798, Saint-Affrique - 20 November 1861) was a French mathematician. ...

 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}

For the first three unit vectors, multiply the elements on the diagonal to the right (e.g. the first diagonal would contain i, a2, and b3). For the last three unit vectors, multiply the elements on the diagonal to the left and then negate the product (e.g. the last diagonal would contain k, a2, and b1). 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).

Although written here in terms of coordinates, it follows from the geometrical definition above that the cross product is invariant under rotations about the axis defined by mathbf{a}timesmathbf{b}, and flips sign under swapping mathbf{a} and mathbf{b}. A sphere rotating around its axis. ...


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. Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...


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 a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...


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: In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = &#8722;A or in component form, if A = (aij): aij = &#8722; aji   for all i and j. ...

mathbf{a} times mathbf{b} = [mathbf{a}]_{times} mathbf{b} = begin{bmatrix}0&-a_3&a_2a_3&0&-a_1-a_2&a_1&0end{bmatrix}begin{bmatrix}b_1b_2b_3end{bmatrix}
mathbf{b} times mathbf{a} = mathbf{b}^T [mathbf{a}]_{times} = begin{bmatrix}b_1&b_2&b_3end{bmatrix}begin{bmatrix}0&-a_3&a_2a_3&0&-a_1-a_2&a_1&0end{bmatrix}

where

[mathbf{a}]_{times} stackrel{rm def}{=} begin{bmatrix}0&-a_3&a_2a_3&0&-a_1-a_2&a_1&0end{bmatrix}

also if mathbf{a} is an result of 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. 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. In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Magnetic field lines shown by iron filings In physics, a magnetic field is a solenoidal vector field in the space surrounding moving electric charges and magnetic dipoles, such as those in electric currents and magnets. ...


This notation is also often much easier to work with, for example, in epipolar geometry. Epipolar geometry refers to the geometry of stereo vision. ...


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.

Lagrange's formula (bac-cab rule) can be easily proven using this notation. This is a well-known and useful formula, a × (b × c) = b(a · c) − c(a · b), which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. ...


The above definition of  [mathbf{a}]_{times} means that there is a one-to-one mapping between the set of  3 times 3 skew-symmetric matrices, also denoted SO(3), and the operation of taking the cross product with some vector  mathbf{a} . In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ...


Index notation

The cross product can alternatively be defined in terms of the Levi-Civita tensor varepsilon_{ijk} The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

 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.


Mnemonic

The word xyzzy can be used to remember the definition of the cross product. Adventure (also known as ADVENT or Colossal Cave) (Crowther & Woods, 1976) was the first computer adventure game. ...


If

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

where:

 mathbf{a} = begin{bmatrix}a_xa_ya_zend{bmatrix}, mathbf{b} = begin{bmatrix}b_xb_yb_zend{bmatrix}, mathbf{c} = begin{bmatrix}c_xc_yc_zend{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, xyzx. 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. Adventure (also known as ADVENT or Colossal Cave) (Crowther & Woods, 1976) was the first computer adventure 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 of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ... For the crossed product in algebra and functional analysis, see crossed product. ... Adventure (also known as ADVENT or Colossal Cave) (Crowther & Woods, 1976) was the first computer adventure game. ...


Applications

The cross product occurs in the formula for the vector operator curl. It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product. A vector operator is a type of differential operator used in vector calculus. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... Lorentz force. ... Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ... This gyroscope remains upright while spinning due to its angular momentum. ...


The cross product can also be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For the journal by ACM SIGGRAPH, see Computer Graphics (Publication). ...


Given a point p and a line through a and b in a plane, all with z coordinate zero, then the z component of (p-a) × (b-a) will be positive or negative, depending on which side of the line p is.


The trick of rewriting a cross product in terms of a matrix multiplication apperars 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. 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 ab as the oriented parallelogram spanned by a and b. We obtain the cross product by taking the Hodge dual of the bivector ab; this can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. In three dimensions only (because only in this case the dual of a vector is a bivector), the result is 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, the cross product of a and b is the vector dual to the bivector ab: 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 ab has relative to the unit bivector; precisely the properties described above. Image File history File links Size of this preview: 684 × 600 pixelsFull resolution (1130 × 991 pixel, file size: 50 KB, MIME type: image/png) The cross product viewed in terms of the exterior product and Hodge dual. ... Image File history File links Size of this preview: 684 × 600 pixelsFull resolution (1130 × 991 pixel, file size: 50 KB, MIME type: image/png) The cross product viewed in terms of the exterior product and Hodge dual. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... This article may be too technical for most readers to understand. ... A bivector is an element of the antisymmetric tensor product of a tangent space with itself. ... In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ...


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. Therefore, for consistency, the other side must also be 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 additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ...


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).


A handedness-free approach is possible using exterior algebra. In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ...


Higher dimensions

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. See seven dimensional cross product for the main article. The nonexistence of 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, the octonions are a nonassociative extension of the quaternions. ... In mathematics, the seven dimensional cross product is a generalization of the three dimensional cross product. ... In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || . || satisfying ||xy|| = ||x|| ||y|| for all x and y in A. While the definition allows normed division algebras to be infinite-dimensional, this, in...


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. The cross product can be interpreted as the wedge product in three dimensions after using Hodge duality to identify 2-vectors with vectors. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... THE TRIVECTOR: the sum of all coolness encapsulated in three rocking houses surrounding victoria park, located in Kingston Ontario Canada. ... In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n&#8722;k-vectors where n = dim V, for 0 &#8804; k &#8804; n. ... THE TRIVECTOR: the sum of all coolness encapsulated in three rocking houses surrounding victoria park, located in Kingston Ontario Canada. ...


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,...,vn,Λ(v1,...,vn)) have a positive orientation with respect to (e1,...,en+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. In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = &#8722;A or in component form, if A = (aij): aij = &#8722; aji   for all i and j. ...


The wedge product and dot product can be combined to form the Clifford product. Clifford algebras are a type of associative algebra in mathematics. ...


In the context of multilinear algebra, it is also 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 tensor of rank n−2. This is a different concept than what is discussed above. In mathematics, multilinear algebra extends the methods of linear algebra. ... Look up Parity in Wiktionary, the free dictionary Parity is a concept of equality of status or functional equivalence. ...


History

In 1843 the Irish mathematical physicist Sir 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. Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance — eponymously named Maxwells equations — including an important modification (extension) of the Ampères... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ...


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. Thus, about forty years after the quaternion product, the dot product and cross product were introduced — to heated opposition. 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. Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and... For other uses, see England (disambiguation). ... Josiah Willard Gibbs (February 11, 1839 New Haven – April 28, 1903 New Haven) was one of the very first American theoretical physicists and chemists. ... Official language(s) English Capital Hartford Largest city Bridgeport Largest metro area Hartford Area  Ranked 48th  - Total 5,543[2] sq mi (14,356 km²)  - Width 70 miles (113 km)  - Length 110 miles (177 km)  - % water 12. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...


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. 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. Hermann Günther Grassmann (April 15, 1809, Stettin – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... William Kingdon Clifford William Kingdon Clifford, FRS (May 4, 1845 - March 3, 1879) was an English mathematician who also wrote a fair bit on philosophy. ... Clifford algebras are a type of associative algebra in mathematics. ...


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 (Wilson 1901, p. 61), a textbook by a former student. Wilson rearranged material from Gibbs's lectures, together with material from publications by Heaviside, Föpps, and Hamilton. He divided vector analysis into three parts: Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ...

"First, that which concerns addition and the scalar and vector products of vectors. 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. The above mentioned triple product expansion (Lagrange's formula) was also included. This article is about mathematics. ... This is a well-known and useful formula, a × (b × c) = b(a · c) − c(a · b), which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. ...


See also

This is a well-known and useful formula, a × (b × c) = b(a · c) − c(a · b), which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. ... This article is about mathematics. ... In mathematics, there are tricks for multiple cross products. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, the Cartesian product is a direct product of sets. ... In its simplest form, multiplication is a quick way of adding identical numbers. ...

References

  • Cajori, Florian (1929), A History Of Mathematical Notations Volume II, Open Court Publishing, pp. 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>

Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ... The Open Court Publishing Company is a publisher with offices in Chicago and La Salle, Illinois. ... Yale University Press is a book publisher founded in 1908. ...

External links


  Results from FactBites:
 
Math Tutorial -- Cross Product (275 words)
As illustrated in figure 11.1, the cross product of two vectors is perpendicular to the plane defined by these vectors.
Note that the magnitude of the cross product is zero when the vectors are parallel or anti-parallel, and maximum when they are perpendicular.
This contrasts with the dot product, which is maximum for parallel vectors and zero for perpendicular vectors.
Vectors, Part 7 (879 words)
The applet shows the cross product of the green vector with the yellow vector (in that order) as the red vector, and all three vectors are also projected onto the xy-plane.
The cross product of v and w is a vector that is perpendicular to both v and w and has length equal to v
The length of the cross product is also the area of the parallelogram determined by the two vectors.
  More results at FactBites »

 
 

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