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Cross product

Cross product

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the cross product, vector product, or Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...

 vector product
is a binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

 on two vectors in three-dimensional space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. It results in a vector which is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

 to both of the vectors being multiplied and normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

 to the plane containing them. The name "cross product" is derived from the cross symbol
×
The multiplication sign is the symbol ×. The symbol is similar to the lowercase letter x but is a more symmetric saltire, and has different uses. It is also known as St...

 "" that is often used to designate this operation; the alternative name "vector product" emphasizes the vector (rather than scalar
Scalar (mathematics)
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....

) nature of the result.
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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the cross product, vector product, or Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...

 vector product
is a binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

 on two vectors in three-dimensional space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. It results in a vector which is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

 to both of the vectors being multiplied and normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

 to the plane containing them. The name "cross product" is derived from the cross symbol
×
The multiplication sign is the symbol ×. The symbol is similar to the lowercase letter x but is a more symmetric saltire, and has different uses. It is also known as St...

 "" that is often used to designate this operation; the alternative name "vector product" emphasizes the vector (rather than scalar
Scalar (mathematics)
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....

) nature of the result. It has many applications in mathematics, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

.

If either of the vectors being multiplied is zero or the vectors are parallel then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular for perpendicular vectors this is a rectangle and the magnitude of the product is the product of their lengths. The cross product is anticommutative
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...

, distributive
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...

 over addition and satisfies the Jacobi identity
Jacobi identity
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. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...

. The space and product form an algebra over a field
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

, which is neither commutative nor associative, but is a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 with the cross product being the Lie bracket.

Like the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

, it depends on the metric
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

 of Euclidean space, but unlike the dot product, it also depends on the choice of orientation
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

 or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...

, or in arbitrary dimensions the exterior product
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

 of vectors can be used with a bivector
Bivector
In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...

 or two-form
Two-form
In linear algebra, a two-form is another term for a bilinear form, typically used in informal discussions, or sometimes to indicate that the bilinear form is skew-symmetric....

 result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.

Definition



The cross product of two vectors a and b is denoted by In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, sometimes the notation is used, though this is avoided in mathematics to avoid confusion with the exterior product.

The cross product a × b is defined as a vector c that is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

 to both a and b, with a direction given by the right-hand rule
Right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....

  and a magnitude equal to the area of the parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

 that the vectors span.

The cross product is defined by the formula


where θ is the measure of the smaller angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 between a and b (0° ≤ θ ≤ 180°), a and b are the magnitudes of vectors a and b (i.e., a = |a| and b = |b|), and n is a unit vector perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

 to the plane containing a and b in the direction given by the right-hand rule as illustrated. If the vectors a and b are parallel (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

The direction of the vector 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 picture on the right). Using this rule implies that the cross-product is anti-commutative
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...

, i.e., b × a = −(a × b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector.

Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction.

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 n. The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...

. See cross product and handedness for more detail.

Coordinate notation


The standard basis
Standard basis
In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...

 vectors i, j, and k satisfy the following equalities:


Together with the skew-symmetry and bilinearity of the product, these three identities are sufficient to determine the cross product of any two vectors. In particular, the following identities can be established:
(the zero vector)

These can be used to compute the product of two general vectors, a = a1i + a2j + a3k and b = b1i + b2j + b3k, by expanding the product using distributivity
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...

 then collecting similar terms:


Or written as column vectors:

Matrix notation


The definition of the cross product can also be represented by the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 of a formal matrix:


This determinant can be computed using Sarrus' rule
Rule of Sarrus
Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus....

 or Cofactor expansion.

Using Sarrus' Rule, it expands to


Using Cofactor expansion along the first row instead, it expands to


which gives the components of the resulting vector directly.

An elegant, alternative derivation is based on an isometric sketch of (x,y,z) axes with the (x,y,z) components of each vector (A,B) drawn parallel to the corresponding axes. The resulting formula for the constituents of the cross product (C = A x B) then appears by inspection, provided we are careful to take all orthogonal cross products in the order A:B (such as Az x Bx). For example, this method yields the magnitude (Cz = AxBy - AyBx) for the vector component along the z-axis and unit vector (k).

Geometric meaning




The magnitude of the cross product can be interpreted as the positive area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 of the parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

 having a and b as sides (see Figure 1):


Indeed, one can also compute the volume V of a parallelepiped
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

 having a, b and c as sides by using a combination of a cross product and a dot product, called scalar triple product (see Figure 2):


Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value. For instance,


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 "perpendicularness" in the same way that the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

 is a measure of "parallelness". Given two 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 opposite is true for the dot product of two unit vectors.

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


The cross product is anticommutative,


distributive over addition,


and compatible with scalar multiplication so that


It is not associative, but satisfies the Jacobi identity
Jacobi identity
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. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...

:


Distributivity, linearity and Jacobi identity show that R3 together with vector addition and the cross product forms a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

, the Lie algebra of the real orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

 in 3 dimensions, SO(3).

The cross product does not obey the cancellation law: a × b = a × c with non-zero a does not imply that b = c. Instead if a × b = a × c:


If neither a nor b - c is zero then from the definition of the cross product the angle between them must be zero and they must be parallel. They are related by a scale factor, so one of b or c can be expressed in terms of the other, for example


for some scalar t.

If a · b = a · c and a × b = a × c, for non-zero vector a, then b = c, as
and

so bc is both parallel and perpendicular to the non-zero vector a, something that is only possible if bc = 0 so they are identical.

From the geometrical definition the cross product is invariant under rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s about the axis defined by a × b. More generally the cross product obeys the following identity under matrix transformations:


where is a 3 by 3 matrix and is the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

 of the inverse

The cross product of two vectors in 3-D always lies in the null space
Null space
In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space...

 of the matrix with the vectors as rows:

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

Differentiation


The product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

 applies to the cross product in a similar manner:
This identity can be easily proved using the matrix multiplication representation.

Triple product expansion


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


It is the signed volume of the parallelepiped
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

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


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


The mnemonic
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

 "BAC minus CAB" is used to remember the order of the vectors in the right hand member. This formula is used in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 to simplify vector calculations. A special case, regarding gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

s and useful in vector calculus, is
where ∇2 is the vector Laplacian
Vector Laplacian
In mathematics and physics, the vector Laplace operator, denoted by \scriptstyle \nabla^2, named after Pierre-Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian...

 operator.

Another identity relates the cross product to the scalar triple product:

Alternative formulation


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


the above given relationship can be rewritten as follows:


Invoking the Pythagorean trigonometric identity
Pythagorean trigonometric identity
The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one the basic relations between the sine and cosine functions, from which all others may be derived.-Statement of...

 one obtains:

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

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

Lagrange's identity


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


where a and b may be n-dimensional vectors. In the case n=3, combining these two equations results in the expression for the magnitude of the cross product in terms of its components:


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

In R3 Lagrange's equation is a special case of the multiplicativity |vw| = |v||w| of the norm in the quaternion algebra.

It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the Binet-Cauchy identity
Binet-Cauchy identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that \biggl\biggl =\biggl\biggl...

:


If a = c and b = d this simplifies to the formula above.

Conversion to matrix multiplication


The vector cross product also can be expressed as the product of a skew-symmetric matrix
Skew-symmetric matrix
In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...

 and a vector:


where superscript T refers to the Transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

 matrix, and [a]× is defined by:


Also, if a is itself a cross product:


then


This result can be generalized to higher dimensions using geometric algebra
Geometric algebra
Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...

. In particular in any dimension bivector
Bivector
In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...

s 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
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...

 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
Epipolar geometry is the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points...

.

From the general properties of the cross product follows immediately that
  and  

and from fact that [a]× is skew-symmetric it follows that


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

The above definition of [a]× means that there is a one-to-one mapping between the set of 3×3 skew-symmetric matrices, also known as the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 of SO(3), and the operation of taking the cross product with some vector a.

Index notation


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

, εijk:
where the indices
Index (mathematics)
The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...

  correspond, as in the previous section, to orthogonal vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention as
in which repeated indices are summed from 1 to 3. Note that this representation is another form of the skew-symmetric representation of the cross product:

In classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

: representing the cross-product with the Levi-Civita symbol can cause mechanical-symmetries to be obvious when physical-systems are isotropic in space. (Quick 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


where:


then:


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 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 above-mentioned 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 help you to remember the correct cross product formula.

If


then:


If we want to obtain the formula for we simply drop the and from the formula, and take the next two components down -


It should be noted that when doing this for 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 , the next two components should be z and x (in that order). While for the next two components should be taken as x and y.


For 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 formula -


We can do this in the same way for and to construct their associated formulas.

Computational geometry


The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

. For example, the winding of polygon (clockwise or anticlockwise) about a point within the polygon (i.e. the centroid or mid-point) 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
Computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...

 of the plane, the cross product is used to determine the sign of the acute angle defined by three points , and . It corresponds to the direction of the cross product of the two coplanar vectors defined by the pairs of points and , i.e., by the sign of the expression . In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a negative angle of rotation around from to , otherwise a positive angle. From another point of view, the sign of tells whether lies to the left or to the right of line .

Mechanics


Moment
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...

 of a force applied at point B around point A is given as:

Other


The cross product occurs in the formula for the vector operator curl.
It is also used to describe the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

 experienced by a moving electrical charge in a magnetic field. The definitions of torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

 and angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

 also involve the cross product.

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

Cross product as an exterior product



The cross product 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
Bivector
In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...

. 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. The cross product is then obtained by taking the Hodge dual
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...

 of the bivector ab, identifying 2-vectors with vectors:


This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented line element – a vector – whereas, for example, in 4 dimensions the Hodge dual of a bivector is two-dimensional – another oriented plane element. So, only in three dimensions is the cross product of a and b 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.

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
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...

. Therefore, for consistency, the other side must also be a pseudovector.

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


So by the above relationships, the unit basis vectors i, j and k of an orthonormal, right-handed (Cartesian) coordinate frame must all be pseudovectors (if a basis of mixed vector types is disallowed, as it normally is) since i × j = k, j × k = i and k × i = j.

Because the cross product may also be a (true) vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a (true) vector and the other one is a pseudovector (e.g., the cross product of two vectors). For instance, a vector triple product involving three (true) vectors is a (true) vector.

A handedness-free approach is possible using exterior algebra
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

.

Generalizations


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

Lie algebra



The cross product can be seen as one of the simplest Lie products,
and is thus generalized by Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

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
Lie theory
Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....

.

For example, the Heisenberg algebra gives another Lie algebra structure on In the basis the product is

Quaternions



The cross product can also be described in terms of quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s, and this is why the letters i, j, k are a convention for the standard basis on . The unit vectors i, j, k correspond to "binary" (180 deg) rotations about their respective axes (Altmann, S. L., 1986, Ch. 12), said rotations being represented by "pure" quaternions (zero scalar part) with unit norms.

For instance, 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 a vector [a1, a2, a3] is represented as the quaternion a1i + a2j + a3k, 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
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

 of the two vectors.

Alternatively and more straightforwardly, using the above identification of the 'purely imaginary' quaternions with , the cross product may be thought of as half of the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

 of two quaternions.

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 such cross products of two vectors in other dimensions is related to the result that the only normed division algebra
Normed division algebra
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 the following property:\|xy\| = \|x\| \|y\| for all x and y in A....

s are the ones with dimension 1, 2, 4, and 8; Hurwitz's theorem.

Wedge product



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. As mentioned above, the cross product can be interpreted as the wedge product in three dimensions after using Hodge duality to identify 2-vectors with vectors.

The wedge product and dot product can be combined to form the Clifford product
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...

.

Multilinear algebra


In the context of multilinear algebra
Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...

, the cross product can be seen as the (1,2)-tensor (a mixed tensor
Mixed tensor
In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript and at least one of the indices will be a superscript ....

, specifically a bilinear map) obtained from the 3-dimensional volume form
Volume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...

,By a volume form one means a function that takes in n vectors and gives out a scalar, the volume of the parallelotope defined by the vectors: This is an n-ary multilinear skew-symmetric form. In the presence of a basis, such as on this is given by the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

, but in an abstract vector space, this is added structure. In terms of G-structures
G-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....

, a volume form is an
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

-structure.
a (0,3)-tensor, by raising an index
Raising and lowering indices
In mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M , one can raise or lower indices: change a type tensor to a tensor or to a tensor...

.

In detail, the 3-dimensional volume form defines a product by taking the determinant of the matrix given by these 3 vectors.
By duality
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

, this is equivalent to a function (fixing any two inputs gives a function by evaluating on the third input) and in the presence of an inner product (such as the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

; more generally, a non-degenerate bilinear form), we have an isomorphism and thus this yields a map 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 " (where the first two vectors are fixed and the last is an input), which defines a function , can be represented uniquely as the dot product with a vector: this vector is the cross product From this perspective, the cross product is defined by the scalar triple product,

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 -tensor.
The most direct generalizations of the cross product are to define either:
  • a -tensor, which takes as input vectors, and gives as output 1 vector – an -ary vector-valued product, or
  • a -tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank n−2 – a binary product with rank n−2 tensor values. One can also define -tensors for other k.


These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

.

The -ary product can be described as follows: given vectors in define their generalized cross product as:
  • perpendicular to the hyperplane defined by the
  • magnitude is the volume of the parallelotope defined by the which can be computed as the Gram determinant of the
  • oriented so that is positively oriented.

This is the unique multilinear, alternating product which evaluates to , and so forth for cyclic permutations of indices.

In coordinates, one can give a formula for this -ary analogue of the cross product in Rn by:


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-1,Λ(v1,...,vn-1)) have a positive orientation
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

 with respect to (e1,...,en). 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 -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.

History


In 1773, Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

 introduced the component form of both the dot and cross products in order to study the tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 in three dimensions. In 1843 the Irish mathematical physicist Sir William Rowan Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

 introduced the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

 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
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

 used Hamilton's quaternion tools to develop his famous electromagnetism equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's 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
William Kingdon Clifford FRS 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 honour, with interesting applications in contemporary mathematical physics...

 published his Elements of Dynamic
Elements of Dynamic
Elements of Dynamic is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies"...

 which was an advanced text for its time. He defined the product of two vectors to have magnitude equal to the area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 of the parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

 of which they are two sides, and direction perpendicular to their plane.

Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...

 in England and Josiah Willard Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...

, a professor at Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...

 in Connecticut
Connecticut
Connecticut is a state in the New England region of the northeastern United States. It is bordered by Rhode Island to the east, Massachusetts to the north, and the state of New York to the west and the south .Connecticut is named for the Connecticut River, the major U.S. river that approximately...

, 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
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

 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
Hermann Grassmann
Hermann Günther Grassmann was a German polymath, renowned in his day as a linguist and now also admired as a mathematician. He was also a physicist, neohumanist, general scholar, and publisher...

 created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford FRS 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 honour, with interesting applications in contemporary mathematical physics...

 combined the algebras of Hamilton and Grassmann to produce Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...

, 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, which began with Gibbs, inspired the name "cross product". Originally it appeared in privately published notes for his students in 1881 as Elements of Vector Analysis. The utility for mechanics was noted by Aleksandr Kotelnikov
Aleksandr Kotelnikov
Aleksandr Petrovich Kotelnikov was a Russian mathematician specializing in geometric algebra.Kotelnikov graduated from Kazan University in 1884 and began teaching at a gymnasium. He began graduate work in mechanics and started instructing at the university in 1893. For his doctorate in 1899 he...

. Gibbs's notation —and the name— later reached a wide audience through Vector Analysis, a textbook by Edwin Bidwell Wilson
Edwin Bidwell Wilson
Edwin Bidwell Wilson was an American mathematician and polymath. He was the sole protégé of Yale's physicist Josiah Willard Gibbs and was mentor to MIT economist Paul Samuelson. He received his AB from Harvard College in 1899 and his PhD from Yale University in 1901, working under Gibbs.E.B...

, 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 product
Triple product
In mathematics, the triple product is a product of three vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product....

s and products of more than three vectors were also examined. The above mentioned triple product expansion was also included.

See also


  • Multiple cross products – Products involving more than three vectors
  • Cartesian product
    Cartesian product
    In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

     – A product of two sets
  • ×
    ×
    The multiplication sign is the symbol ×. The symbol is similar to the lowercase letter x but is a more symmetric saltire, and has different uses. It is also known as St...

     (the symbol)
  • Bivector
    Bivector
    In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...

  • Pseudovector
    Pseudovector
    In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...


External links