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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a
binary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
on two
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
s in a three-dimensional
oriented is non-orientable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...
Euclidean vector space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called p ...
(named here E), and is denoted by the symbol \times. Given two
linearly independent vectors In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said t ...
and , the cross product, (read "a cross b"), is a vector that is
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

perpendicular
to both and , and thus normal to the plane containing them. It has many applications in mathematics,
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specializ ...

engineering
, and
computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generating algorithms, Profilin ...
. It should not be confused with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
(projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are ''not'' linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is
anticommutative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
(that is, ) and is distributive over addition (that is, ). The space E together with the cross product is an algebra over the real numbers, which is neither
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
nor
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, but is a
Lie algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
with the cross product being the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightar ...
. Like the dot product, it depends on the
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
of
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, but unlike the dot product, it also depends on a choice of
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
(or "
handedness In human biology Human biology is an interdisciplinary area of academic study that examines humans through the influences and interplay of many diverse fields such as human genetics, genetics, human evolution, evolution, human physiology, physio ...

handedness
") of the space (it's why an oriented space is needed). In connection with the cross product, the
exterior product In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
of vectors can be used in arbitrary dimensions (with a
bivectorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

bivector
or
2-form In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
result) and is independent of the orientation of the space. The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in dimensions, take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. (See § Generalizations, below, for other dimensions.)


Definition

The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by . In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
and
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...
, 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 In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

perpendicular
(orthogonal) to both a and b, with a direction given by the
right-hand rule In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

right-hand rule
and a magnitude equal to the area of the
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
that the vectors span. The cross product is defined by the formula :\mathbf \times \mathbf = \left\, \mathbf \right\, \left\, \mathbf \right\, \sin (\theta) \ \mathbf where: * ''θ'' is the
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

angle
between a and b in the plane containing them (hence, it is between 0° and 180°) * ‖a‖ and ‖b‖ are the magnitudes of vectors a and b * and n is a
unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

perpendicular
to the plane containing a and b, in the direction given by the right-hand rule (illustrated). 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
0.


Direction

By convention, the direction of the vector 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 n is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is
anti-commutative In mathematics, anticommutativity is a specific property of some non-commutative Operation (mathematics), operations. In mathematical physics, where symmetry (physics), symmetry is of central importance, these operations are mostly called antisymmet ...
; 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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
of the space (as explicit in the definition above), the cross product of two vectors is not a "true" vector, but a ''pseudovector''. See for more detail.


Names

In 1881,
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in tr ...

Josiah Willard Gibbs
, and independently
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English autodidactic Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education without the guidance of masters (such as teach ...
, introduced both the dot product and the cross product using a period () and an "x" (), respectively, to denote them.''A History of Vector Analysis'' by Michael J. Crowe
Math. UC Davis
In 1877, to emphasize the fact that the result of a dot product is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
while the result of a cross product is a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
,
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

William Kingdon Clifford
coined the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature. Both the cross notation () and the name cross product were possibly inspired by the fact that each scalar component of is computed by multiplying non-corresponding components of a and b. Conversely, a dot product involves multiplications between corresponding components of a and b. As explained
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
, the cross product can be expressed in the form of a determinant of a special matrix. According to Sarrus's rule, this involves multiplications between matrix elements identified by crossed diagonals.


Computing


Coordinate notation

If (i, j,k) is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities :\begin \mathbf&\times\mathbf &&= \mathbf\\ \mathbf&\times\mathbf &&= \mathbf\\ \mathbf&\times\mathbf &&= \mathbf \end which imply, by the
anticommutativity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of the cross product, that :\begin \mathbf&\times\mathbf &&= -\mathbf\\ \mathbf&\times\mathbf &&= -\mathbf\\ \mathbf&\times\mathbf &&= -\mathbf \end The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that :\mathbf\times\mathbf = \mathbf\times\mathbf = \mathbf\times\mathbf = \mathbf (the
zero vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
). These equalities, together with the
distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and
linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...

linearity
of the cross product (though neither follows easily from the definition given above), are sufficient to determine the cross product of any two vectors a and b. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors: :\begin \mathbf &= a_1\mathbf &&+ a_2\mathbf &&+ a_3\mathbf \\ \mathbf &= b_1\mathbf &&+ b_2\mathbf &&+ b_3\mathbf \end Their cross product can be expanded using distributivity: : \begin \mathbf\times\mathbf = &(a_1\mathbf + a_2\mathbf + a_3\mathbf) \times (b_1\mathbf + b_2\mathbf + b_3\mathbf)\\ = &a_1b_1(\mathbf \times \mathbf) + a_1b_2(\mathbf \times \mathbf) + a_1b_3(\mathbf \times \mathbf) + \\ &a_2b_1(\mathbf \times \mathbf) + a_2b_2(\mathbf \times \mathbf) + a_2b_3(\mathbf \times \mathbf) + \\ &a_3b_1(\mathbf \times \mathbf) + a_3b_2(\mathbf \times \mathbf) + a_3b_3(\mathbf \times \mathbf)\\ \end This can be interpreted as the decomposition of into the sum of nine simpler cross products involving vectors aligned with i, j, or k. Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using the above-mentioned equalities and collecting similar terms, we obtain: :\begin \mathbf\times\mathbf = &\quad\ a_1b_1\mathbf + a_1b_2\mathbf - a_1b_3\mathbf \\ &- a_2b_1\mathbf + a_2b_2\mathbf + a_2b_3\mathbf \\ &+ a_3b_1\mathbf\ - a_3b_2\mathbf\ + a_3b_3\mathbf \\ = &(a_2b_3 - a_3b_2)\mathbf + (a_3b_1 - a_1b_3)\mathbf + (a_1b_2 - a_2b_1)\mathbf\\ \end meaning that the three scalar components of the resulting vector s = ''s''1i + ''s''2j + ''s''3k = are :\begin s_1 &= a_2b_3-a_3b_2\\ s_2 &= a_3b_1-a_1b_3\\ s_3 &= a_1b_2-a_2b_1 \end Using
column vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
s, we can represent the same result as follows: :\begins_1\\s_2\\s_3\end=\begina_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end


Matrix notation

The cross product can also be expressed as the
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for forma ...
determinant:Here, "formal" means that this notation has the form of a determinant, but does not strictly adhere to the definition; it is a mnemonic used to remember the expansion of the cross product. :\mathbf = \begin \mathbf&\mathbf&\mathbf\\ a_1&a_2&a_3\\ b_1&b_2&b_3\\ \end This determinant can be computed using Sarrus's rule or
cofactor expansion In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineerin ...
. Using Sarrus's rule, it expands to :\begin \mathbf &=(a_2b_3\mathbf+a_3b_1\mathbf+a_1b_2\mathbf) - (a_3b_2\mathbf+a_1b_3\mathbf+a_2b_1\mathbf)\\ &=(a_2b_3 - a_3b_2)\mathbf +(a_3b_1 - a_1b_3)\mathbf +(a_1b_2 - a_2b_1)\mathbf. \end Using cofactor expansion along the first row instead, it expands to :\begin \mathbf &= \begin a_2&a_3\\ b_2&b_3 \end\mathbf - \begin a_1&a_3\\ b_1&b_3 \end\mathbf + \begin a_1&a_2\\ b_1&b_2 \end\mathbf \\ &=(a_2b_3 - a_3b_2)\mathbf -(a_1b_3 - a_3b_1)\mathbf +(a_1b_2 - a_2b_1)\mathbf, \end which gives the components of the resulting vector directly.


Using Levi-Civita tensors

* In any basis, the cross-product a \times b is given by the tensorial formula E_a^ib^j where E_ is the covariant
Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italians, Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made s ...
tensor (we note the position of the indices). That corresponds to the intrinsic formula given
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV Here TV is an America ...
. * In an orthonormal basis having the same orientation as the space, a \times b is given by the pseudo-tensorial formula \varepsilon_a^ib^j where \varepsilon_ is the Levi-Civita symbol (which is a pseudo-tensor). That’s the formula used for everyday physics but it works only for this special choice of basis. * In any orthonormal basis, a \times b is given by the pseudo-tensorial formula (-1)^B\varepsilon_a^ib^j where (-1)^B = \pm 1 indicates whether the basis has the same orientation as the space or not. The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis.


Properties


Geometric meaning

The
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of the cross product can be interpreted as the positive
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
of the
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
having a and b as sides (see Figure 1): : \left\, \mathbf \times \mathbf \right\, = \left\, \mathbf \right\, \left\, \mathbf \right\, , \sin \theta , . Indeed, one can also compute the volume ''V'' of a
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

parallelepiped
having a, b and c as edges by using a combination of a cross product and a dot product, called
scalar triple product In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

scalar triple product
(see Figure 2): : \mathbf\cdot(\mathbf\times \mathbf)= \mathbf\cdot(\mathbf\times \mathbf)= \mathbf\cdot(\mathbf\times \mathbf). Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value: :V = , \mathbf \cdot (\mathbf \times \mathbf), . Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of ''perpendicularity'' in the same way that the dot product is a measure of ''parallelism''. Given two
unit vectors In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

unit vectors
, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel. Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive).


Algebraic properties

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: :\mathbf \times \mathbf = \mathbf. The cross product is
anticommutative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, :\mathbf \times \mathbf = -(\mathbf \times \mathbf), distributive over addition, : \mathbf \times (\mathbf + \mathbf) = (\mathbf \times \mathbf) + (\mathbf \times \mathbf), and compatible with scalar multiplication so that :(r\,\mathbf) \times \mathbf = \mathbf \times (r\,\mathbf) = r\,(\mathbf \times \mathbf). It is not
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, but satisfies the
Jacobi identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
: :\mathbf \times (\mathbf \times \mathbf) + \mathbf \times (\mathbf \times \mathbf) + \mathbf \times (\mathbf \times \mathbf) = \mathbf. Distributivity, linearity and Jacobi identity show that the R3
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
together with vector addition and the cross product forms a
Lie algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, the Lie algebra of the real
orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
in 3 dimensions,
SO(3) In classical mechanics, mechanics and geometry, the 3D rotation group, often denoted special orthogonal group, SO(3), is the group (mathematics), group of all rotations about the origin (mathematics), origin of three-dimensional space, three-dimensi ...
. The cross product does not obey the
cancellation law In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma (algebra), magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always imp ...
; that is, with does not imply , but only that: : \begin \mathbf &= (\mathbf \times \mathbf) - (\mathbf \times \mathbf)\\ &= \mathbf \times (\mathbf - \mathbf).\\ \end 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: :\mathbf = \mathbf + t\,\mathbf, for some scalar ''t''. If, in addition to and as above, it is the case that then :\begin \mathbf \times (\mathbf - \mathbf) &= \mathbf \\ \mathbf \cdot (\mathbf - \mathbf) &= 0, \end 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 A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
about the axis defined by . In formulae: :(R\mathbf) \times (R\mathbf) = R(\mathbf \times \mathbf), where R is a
rotation matrix In , a rotation matrix is a that is used to perform a in . For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through ...
with \det(R)=1. More generally, the cross product obeys the following identity under
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
transformations: :(M\mathbf) \times (M\mathbf) = (\det M) \left(M^\right)^\mathrm(\mathbf \times \mathbf) = \operatorname M (\mathbf \times \mathbf) where M is a 3-by-3
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
and \left(M^\right)^\mathrm is the
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

transpose
of the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
and \operatorname is the cofactor matrix. It can be readily seen how this formula reduces to the former one if M is a rotation matrix. The cross product of two vectors lies in the
null space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the matrix with the vectors as rows: :\mathbf \times \mathbf \in NS\left(\begin\mathbf \\ \mathbf\end\right). For the sum of two cross products, the following identity holds: :\mathbf \times \mathbf + \mathbf \times \mathbf = (\mathbf - \mathbf) \times (\mathbf - \mathbf) + \mathbf \times \mathbf + \mathbf \times \mathbf.


Differentiation

The
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more Functions (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...
of differential calculus applies to any bilinear operation, and therefore also to the cross product: :\frac(\mathbf \times \mathbf) = \frac \times \mathbf + \mathbf \times \frac , 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

scalar triple product
of three vectors is defined as :\mathbf \cdot (\mathbf \times \mathbf), It is the signed volume of the
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

parallelepiped
with edges a, b and c and as such the vectors can be used in any order that's an
even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ord ...
of the above ordering. The following therefore are equal: :\mathbf \cdot (\mathbf \times \mathbf) = \mathbf \cdot (\mathbf \times \mathbf) = \mathbf \cdot (\mathbf \times \mathbf), 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 :\mathbf \times (\mathbf \times \mathbf) = \mathbf(\mathbf \cdot \mathbf) - \mathbf(\mathbf \cdot \mathbf). The
mnemonic A mnemonic () device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory Memory is the faculty of the by which or is , stored, and retrieved when needed. It is 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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
to simplify vector calculations. A special case, regarding
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

gradient
s and useful in
vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in ...
, is :\begin \nabla \times (\nabla \times \mathbf) &= \nabla (\nabla \cdot \mathbf ) - (\nabla \cdot \nabla) \mathbf \\ &= \nabla (\nabla \cdot \mathbf ) - \nabla^2 \mathbf,\\ \end where ∇2 is the vector Laplacian operator. Other identities relate the cross product to the scalar triple product: :\begin (\mathbf\times \mathbf)\times (\mathbf\times \mathbf) &= (\mathbf\cdot(\mathbf\times \mathbf)) \mathbf \\ (\mathbf\times \mathbf)\cdot(\mathbf\times \mathbf) &= \mathbf^\mathrm \left( \left( \mathbf^\mathrm \mathbf\right)I - \mathbf \mathbf^\mathrm \right) \mathbf\\ &= (\mathbf\cdot \mathbf)(\mathbf\cdot \mathbf)-(\mathbf\cdot \mathbf) (\mathbf\cdot \mathbf) \end where ''I'' is the identity matrix.


Alternative formulation

The cross product and the dot product are related by: : \left\, \mathbf \times \mathbf \right\, ^2 = \left\, \mathbf\right\, ^2 \left\, \mathbf\right\, ^2 - (\mathbf \cdot \mathbf)^2 . The right-hand side is the
Gram determinant In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by G_ = \left\langle v_i, v_j \right\rangle., p.441, Theo ...
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: : \mathbf = \left\, \mathbf a \right\, \left\, \mathbf b \right\, \cos \theta , the above given relationship can be rewritten as follows: : \left\, \mathbf \right\, ^2 = \left\, \mathbf \right\, ^2 \left\, \mathbf\right \, ^2 \left(1-\cos^2 \theta \right) . Invoking the
Pythagorean trigonometric identity The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an List of trigonometric identities, identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the trigonometric identity#Angl ...
one obtains: : \left\, \mathbf \times \mathbf \right\, = \left\, \mathbf \right\, \left\, \mathbf \right\, \left, \sin \theta \ , 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 A definition is a statement of the meaning of a term (a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical ...
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.


Lagrange's identity

The relation: : \left\, \mathbf \times \mathbf \right\, ^2 \equiv \det \begin \mathbf \cdot \mathbf & \mathbf \cdot \mathbf \\ \mathbf \cdot \mathbf & \mathbf \cdot \mathbf\\ \end \equiv \left\, \mathbf \right\, ^2 \left\, \mathbf \right\, ^2 - (\mathbf \cdot \mathbf)^2 . can be compared with another relation involving the right-hand side, namely
Lagrange's identity In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
expressed as: : \sum_ \left( a_ib_j - a_jb_i \right)^2 \equiv \left\, \mathbf a \right\, ^2 \left\, \mathbf b \right\, ^2 - ( \mathbf )^2\ , where a and b may be ''n''-dimensional vectors. This also shows that the
Riemannian volume formIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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: :\begin &\left\, \mathbf \times \mathbf\right\, ^2 \equiv \sum_\left(a_ib_j - a_jb_i \right)^2 \\ \equiv &\left(a_1b_2 - b_1a_2\right)^2 + \left(a_2b_3 - a_3b_2\right)^2 + \left(a_3b_1 - a_1b_3\right)^2 \ . \end The same result is found directly using the components of the cross product found from: :\mathbf \times \mathbf \equiv \det \begin \hat\mathbf & \hat\mathbf & \hat\mathbf \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end. In R3, Lagrange's equation is a special case of the multiplicativity of the norm in the
quaternion algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. 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:by : (\mathbf \times \mathbf) \cdot (\mathbf \times \mathbf) \equiv (\mathbf \cdot \mathbf)(\mathbf \cdot \mathbf) - (\mathbf \cdot \mathbf)(\mathbf \cdot \mathbf). If and this simplifies to the formula above.


Infinitesimal generators of rotations

The cross product conveniently describes the infinitesimal generators of
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
s in R3. Specifically, if n is a unit vector in R3 and ''R''(''φ'', n) denotes a rotation about the axis through the origin specified by n, with angle φ (measured in radians, counterclockwise when viewed from the tip of n), then :\left. \_ R(\phi,\boldsymbol) \boldsymbol = \boldsymbol \times \boldsymbol for every vector x in R3. The cross product with n therefore describes the infinitesimal generator of the rotations about n. These infinitesimal generators form the
Lie algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
so(3) of the
rotation group SO(3) In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...
, and we obtain the result that the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3).


Alternative ways to compute


Conversion to matrix multiplication

The vector cross product also can be expressed as the product of a
skew-symmetric matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
and a vector: :\begin \mathbf \times \mathbf = mathbf \mathbf &= \begin\,0&\!-a_3&\,\,a_2\\ \,\,a_3&0&\!-a_1\\-a_2&\,\,a_1&\,0\end\beginb_1\\b_2\\b_3\end \\ \mathbf \times \mathbf = ^\mathrm \mathbf &= \begin\,0&\,\,b_3&\!-b_2\\ -b_3&0&\,\,b_1\\\,\,b_2&\!-b_1&\,0\end\begina_1\\a_2\\a_3\end, \end where superscript refers to the
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

transpose
operation, and ''asub>× is defined by: : mathbf \stackrel \begin\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end. The columns ''asub>×,i of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with
unit vectors In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

unit vectors
. That is, : mathbf = \mathbf \times \mathbf, \; i\in \ or : mathbf = \sum_^3\left(\mathbf \times \mathbf\right)\otimes\mathbf, where \otimes is the
outer product In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
operator. Also, if a is itself expressed as a cross product: :\mathbf = \mathbf \times \mathbf then : mathbf = \mathbf\mathbf^\mathrm - \mathbf\mathbf^\mathrm . : This result can be generalized to higher dimensions using
geometric algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * 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 notation is also often much easier to work with, for example, in
epipolar geometry Epipolar geometry is the geometry of stereo vision#Computer stereo vision, stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the ...

epipolar geometry
. From the general properties of the cross product follows immediately that : mathbf \, \mathbf = \mathbf   and   \mathbf^\mathrm T \, mathbf = \mathbf and from fact that ''asub>× is skew-symmetric it follows that :\mathbf^\mathrm T \, mathbf \, \mathbf = 0. The above-mentioned triple product expansion (bac–cab rule) can be easily proven using this notation. As mentioned above, the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3), whose elements can be identified with the 3×3 skew-symmetric matrices. The map a → ''asub>× provides an isomorphism between R3 and so(3). Under this map, the cross product of 3-vectors corresponds to the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of 3x3 skew-symmetric matrices. :


Index notation for tensors

The cross product can alternatively be defined in terms of the
Levi-Civita tensor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
''Eijk'' and a dot product ''ηmi'', which are useful in converting vector notation for tensor applications: :\mathbf = \mathbf \Leftrightarrow\ c^m = \sum_^3 \sum_^3 \sum_^3 \eta^ E_ a^j b^k where the indices i,j,k correspond to vector components. This characterization of the cross product is often expressed more compactly using the
Einstein summation convention In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
as :\mathbf = \mathbf \Leftrightarrow\ c^m = \eta^ E_ a^j b^k in which repeated indices are summed over the values 1 to 3. In a positively-oriented orthonormal basis ''ηmi'' = δ''mi'' (the
Kronecker delta In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
) and E_ = \varepsilon_ (the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the natural numbers , for som ...
). In that case, this representation is another form of the skew-symmetric representation of the cross product: : varepsilon_ a^j= mathbf\times. In classical mechanics: representing the cross product by using the Levi-Civita symbol can cause mechanical symmetries to be obvious when physical systems are isotropic. (An example: consider a particle in a Hooke's Law potential in three-space, free to oscillate in three dimensions; none of these dimensions are "special" in any sense, so symmetries lie in the cross-product-represented angular momentum, which are made clear by the abovementioned Levi-Civita representation).


Mnemonic

The word "xyzzy" can be used to remember the definition of the cross product. If :\mathbf = \mathbf \times \mathbf where: : \mathbf = \begina_x\\a_y\\a_z\end,\ \mathbf = \beginb_x\\b_y\\b_z\end,\ \mathbf = \beginc_x\\c_y\\c_z\end 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. The second and third equations can be obtained from the first by simply vertically rotating the subscripts, . The problem, of course, is how to remember the first equation, and two options are available for this purpose: either to remember the relevant two diagonals of Sarrus's scheme (those containing ''i''), or to remember the xyzzy sequence. Since the first diagonal in Sarrus's scheme is just the main diagonal of the cross product#Matrix notation, above-mentioned 3×3 matrix, the first three letters of the word xyzzy can be very easily remembered.


Cross visualization

Similarly to the mnemonic device above, a "cross" or X can be visualized between the two vectors in the equation. This may be helpful for remembering the correct cross product formula. If :\mathbf = \mathbf \times \mathbf then: : \mathbf = \beginb_x\\b_y\\b_z\end \times \beginc_x\\c_y\\c_z\end. If we want to obtain the formula for a_x we simply drop the b_x and c_x from the formula, and take the next two components down: : a_x = \beginb_y\\b_z\end \times \beginc_y\\c_z\end. When doing this for a_y the next two elements down should "wrap around" the matrix so that after the z component comes the x component. For clarity, when performing this operation for a_y, the next two components should be z and x (in that order). While for a_z the next two components should be taken as x and y. : a_y = \beginb_z\\b_x\end \times \beginc_z\\c_x\end,\ a_z = \beginb_x\\b_y\end \times \beginc_x\\c_y\end For a_x then, if we visualize the cross operator as pointing from an element on the left to an element on the right, we can take the first element on the left and simply multiply by the element that the cross points to in the right hand matrix. We then subtract the next element down on the left, multiplied by the element that the cross points to here as well. This results in our a_x formula – :a_x = b_y c_z - b_z c_y. We can do this in the same way for a_y and a_z to construct their associated formulas.


Applications

The cross product has applications in various contexts. For example, it is used in computational geometry, physics and engineering. A non-exhaustive list of examples follows.


Computational geometry

The cross product appears in the calculation of the distance of two Skew lines#Distance, skew lines (lines not in the same plane) from each other in three-dimensional space. The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle. 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). It corresponds to the direction (upward or downward) of the cross product of the two coplanar vector (geometry), vectors defined by the two pairs of points (p_1, p_2) and (p_1, p_3). The sign of the acute angle is the sign of the expression : P = (x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1), which is the signed length of the cross product of the two vectors. In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around p_1 from p_2 to p_3, otherwise a negative 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. The cross product is used in calculating the volume of a polyhedron such as a tetrahedron#Volume, tetrahedron or parallelepiped#Volume, parallelepiped.


Angular momentum and torque

The angular momentum of a particle about a given origin is defined as: : \mathbf = \mathbf \times \mathbf, where is the position vector of the particle relative to the origin, is the linear momentum of the particle. In the same way, the Moment (physics), moment of a force applied at point B around point A is given as: : \mathbf_\mathrm = \mathbf_\mathrm \times \mathbf_\mathrm\, In mechanics the ''moment of a force'' is also called ''torque'' and written as \mathbf Since position linear momentum and force are all ''true'' vectors, both the angular momentum and the moment of a force are ''pseudovectors'' or ''axial vectors''.


Rigid body

The cross product frequently appears in the description of rigid motions. Two points ''P'' and ''Q'' on a rigid body can be related by: : \mathbf_P - \mathbf_Q = \boldsymbol\omega \times \left( \mathbf_P - \mathbf_Q \right)\, where \mathbf is the point's position, \mathbf is its velocity and \boldsymbol\omega is the body's angular velocity. Since position \mathbf and velocity \mathbf are ''true'' vectors, the angular velocity \boldsymbol\omega is a ''pseudovector'' or ''axial vector''.


Lorentz force

The cross product is used to describe the Lorentz force experienced by a moving electric charge : \mathbf = q_e \left( \mathbf+ \mathbf \times \mathbf \right) Since velocity force and electric field are all ''true'' vectors, the magnetic field is a ''pseudovector''.


Other

In
vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in ...
, the cross product is used to define the formula for the vector operator Curl (mathematics), curl. The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar geometry, epipolar and multi-view geometry, in particular when deriving matching constraints.


As an external product

The cross product can be defined in terms of the exterior product. It can be generalized to an Cross product#External product, external product in other than three dimensions. This view allows for 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 p-vector, 2-vectors to vectors: :a \times b = \star (a \wedge b) \,. This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented one-dimensional element – a vector – whereas, for example, in four dimensions the Hodge dual of a bivector is two-dimensional – a bivector. So, only in three dimensions can a vector cross product of ''a'' and ''b'' be defined as the vector dual to the bivector : 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 has relative to the unit bivector; precisely the properties described above.


Handedness


Consistency

When physics laws are written as equations, it is possible to make an arbitrary choice of the coordinate system, including handedness. 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 polar vectors, one must take into account that the result is an Pseudovector, axial vector. Therefore, for consistency, the other side must also be an axial vector. More generally, the result of a cross product may be either a polar vector or an axial vector, depending on the type of its operands (polar vectors or axial vectors). Namely, polar vectors and axial vectors are interrelated in the following ways under application of the cross product: * polar vector × polar vector = axial vector * axial vector × axial vector = axial vector * polar vector × axial vector = polar vector * axial vector × polar vector = polar vector or symbolically * polar × polar = axial * axial × axial = axial * polar × axial = polar * axial × polar = polar Because the cross product may also be a polar 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 polar vector and the other one is an axial vector (e.g., the cross product of two polar vectors). For instance, a vector triple product involving three polar vectors is a polar vector. A handedness-free approach is possible using exterior algebra.


The paradox of the orthonormal basis

Let (i, j,k) be an orthonormal basis. The vectors i, j and k don't depend on the orientation of the space. They can even be defined in the absence of any orientation. They can not therefore be axial vectors. But if i and j are polar vectors then k is an axial vector for i × j = k or j × i = k. This is a paradox. "Axial" and "polar" are ''physical'' qualifiers for ''physical'' vectors; that is, vectors which represent ''physical'' quantities such as the velocity or the magnetic field. The vectors i, j and k are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector. There is no contradiction.


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 algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, 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 \mathbf^3. In the basis \, the product is [x,y]=z, [x,z]=[y,z]=0.


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 (normed division algebras), 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 p-vector, 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 algebra, 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 bivector, 2-vector (not the 1-vector) Graded vector space, subalgebra of the three-dimensional geometric algebra, where \mathbf = \mathbf \mathbf, \mathbf = \mathbf \mathbf, and \mathbf = \mathbf \mathbf, the cross product corresponds exactly to the geometric algebra#Extensions of the inner and exterior products, commutator product in geometric algebra and both use the same symbol \times. The commutator product is defined for 2-vectors A and B in geometric algebra as: : A \times B = \tfrac(AB - BA) where AB is the geometric product. The commutator product could be generalised to arbitrary multivector#Geometric algebra, multivectors in three dimensions, which results in a multivector consisting of only elements of Graded vector space, grades 1 (1-vectors/Cross product#Cross product and handedness, 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 Geometric algebra#Extensions of the inner and exterior products, 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 Sign (mathematics)#Sign of a direction, 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 Cross product#Lie algebra, 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,By a volume form one means a function that takes in ''n'' vectors and gives out a scalar, the volume of the Parallelepiped#Parallelotope, parallelotope defined by the vectors: V\times \cdots \times V \to \mathbf. This is an ''n''-ary multilinear skew-symmetric form. In the presence of a basis, such as on \mathbf^n, this is given by the determinant, but in an abstract vector space, this is added structure. In terms of G-structure, ''G''-structures, a volume form is an Special linear group, SL-structure. a (0,3)-tensor, by Raising and lowering indices, raising an index. In detail, the 3-dimensional volume form defines a product V \times V \times V \to \mathbf, by taking the determinant of the matrix given by these 3 vectors. By Dual space, duality, this is equivalent to a function V \times V \to V^*, (fixing any two inputs gives a function V \to \mathbf 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 V \to V^*, and thus this yields a map V \times V \to V, 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 (a,b,-)" (where the first two vectors are fixed and the last is an input), which defines a function V \to \mathbf, can be ''represented'' uniquely as the dot product with a vector: this vector is the cross product a \times b. From this perspective, the cross product is ''defined'' by the
scalar triple product In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

scalar triple product
, \mathrm(a,b,c) = (a\times b)\cdot c. 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 (0,n)-tensor. The most direct generalizations of the cross product are to define either: * a (1,n-1)-tensor, which takes as input n-1 vectors, and gives as output 1 vector – an (n-1)-ary vector-valued product, or * a (n-2,2)-tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank – a binary product with rank tensor values. One can also define (k,n-k)-tensors for other ''k''. These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity (physics), parity. The (n-1)-ary product can be described as follows: given n-1 vectors v_1,\dots,v_ in \mathbf^n, define their generalized cross product v_n = v_1 \times \cdots \times v_ as: * perpendicular to the hyperplane defined by the v_i, * magnitude is the volume of the parallelotope defined by the v_i, which can be computed as the Gram determinant of the v_i, * oriented so that v_1,\dots,v_n is positively oriented. This is the unique multilinear, alternating product which evaluates to e_1 \times \cdots \times e_ = e_n, e_2 \times \cdots \times e_n = e_1, and so forth for cyclic permutations of indices. In coordinates, one can give a formula for this (n-1)-ary analogue of the cross product in R''n'' by: :\bigwedge_^\mathbf_i = \begin v_1^1 &\cdots &v_1^\\ \vdots &\ddots &\vdots\\ v_^1 & \cdots &v_^\\ \mathbf_1 &\cdots &\mathbf_ \end. 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''−1, Λv''i'') have a positive
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
with respect to (e1, ..., e''n''). 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 (n-1)-ary form enjoys many of the same properties as the vector cross product: it is alternating form, 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.


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. 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 R3, their quaternion product can be summarized as . James Clerk Maxwell used Hamilton's quaternion tools to develop his famous Maxwell's equations, electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education. In 1878
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

William Kingdon Clifford
published his Elements of Dynamic which was an advanced text for its time. He defined the product of two vectors to have magnitude equal to the
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
of the
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
of which they are two sides, and direction perpendicular to their plane.
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English autodidactic Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education without the guidance of masters (such as teach ...
and
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in tr ...

Josiah Willard Gibbs
also 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. 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. 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. 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. The cross notation and the name "cross product" began with Gibbs. Originally they appeared in privately published notes for his students in 1881 as ''Elements of Vector Analysis''. The utility for mechanics was noted by Aleksandr Kotelnikov. Gibbs's notation and the name "cross product" later reached a wide audience through Vector Analysis, a textbook by Edwin Bidwell Wilson, 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: 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 was also included.


See also

* Cartesian product – A product of two sets * Geometric algebra#Rotating systems, 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. * *


External links

*
A quick geometrical derivation and interpretation of cross products


created at Syracuse University – (requires Java (programming language), 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 {{DEFAULTSORT:Cross Product Bilinear operators Operations on vectors Analytic geometry