In
mathematicsMathematics 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
GibbsJosiah 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 operationIn 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 threedimensional
spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. It results in a vector which is
perpendicularIn 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
normalA surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a nonflat 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 symbolThe 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
scalarIn 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,
engineeringEngineering 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
physicsPhysics 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
anticommutativeIn 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...
,
distributiveIn 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 identityIn 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 fieldIn 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 algebraIn 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 productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
, it depends on the
metricIn 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 3dimensional Euclidean space...
of Euclidean space, but unlike the dot product, it also depends on the choice of
orientationIn 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 lefthanded or righthanded. 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
pseudovectorIn 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 productIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higherdimensional analogs...
of vectors can be used with a
bivectorIn mathematics, a bivector or 2vector 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
twoformIn linear algebra, a twoform is another term for a bilinear form, typically used in informal discussions, or sometimes to indicate that the bilinear form is skewsymmetric....
result. Also, using the orientation and metric structure just as for the traditional 3dimensional 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 nontrivial 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
physicsPhysics 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
perpendicularIn 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
righthand ruleIn mathematics and physics, the righthand 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
parallelogramIn 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
angleIn 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 perpendicularIn 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 righthand 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 righthand 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 crossproduct is
anticommutativeIn 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 lefthanded coordinate system is used, the direction of the vector
n is given by the lefthand 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 righthanded to a lefthanded coordinate system), should not change the direction of
n. The problem is clarified by realizing that the crossproduct of two vectors is not a (true) vector, but rather a
pseudovectorIn 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 basisIn 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 skewsymmetry 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 =
a_{1}i +
a_{2}j +
a_{3}k and
b =
b_{1}i +
b_{2}j +
b_{3}k, by expanding the product using
distributivityIn 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
determinantIn 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' ruleSarrus' 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 zaxis and unit vector (k).
Geometric meaning
The magnitude of the cross product can be interpreted as the positive
areaArea is a quantity that expresses the extent of a twodimensional 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
parallelogramIn 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
parallelepipedIn geometry, a parallelepiped is a threedimensional 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 productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength 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 identityIn 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
R^{3} together with vector addition and the cross product forms a
Lie algebraIn 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 groupIn 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 nonzero
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 nonzero vector
a, then
b =
c, as
and
so
b −
c is both parallel and perpendicular to the nonzero vector
a, something that is only possible if
b −
c =
0 so they are identical.
From the geometrical definition the cross product is invariant under
rotationA rotation is a circular movement of an object around a center of rotation. A threedimensional 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
transposeIn 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 3D always lies in the
null spaceIn 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 ndimensional Euclidean space...
of the matrix with the vectors as rows:
For the sum of two cross products, the following identity holds:
Differentiation
The
product ruleIn 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
parallelepipedIn geometry, a parallelepiped is a threedimensional 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
mnemonicA 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
physicsPhysics 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
gradientIn 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 LaplacianIn mathematics and physics, the vector Laplace operator, denoted by \scriptstyle \nabla^2, named after PierreSimon 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 righthand 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 identityThe Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sumofangles 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 righthand side, namely
Lagrange's identity expressed as:
where
a and
b may be
ndimensional 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 crossproduct found from:
In
R^{3} 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
BinetCauchy identityIn algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and AugustinLouis 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
skewsymmetric matrixIn mathematics, and in particular linear algebra, a skewsymmetric 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
TransposeIn 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 algebraGeometric 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
bivectorIn mathematics, a bivector or 2vector 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 skewsymmetric matrices, so the product between a skewsymmetric matrix and vector is equivalent to the grade1 part of the product of a bivector and vector. In three dimensions bivectors are
dualIn 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 finitedimensional 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 geometryEpipolar 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 skewsymmetric it follows that
The abovementioned triple product expansion (baccab rule) can be easily proven using this notation.
The above definition of [
a]
_{×} means that there is a onetoone mapping between the set of 3×3 skewsymmetric matrices, also known as the
Lie algebraIn 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
LeviCivita symbolThe LeviCivita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
, ε
_{ijk}:
where the
indicesThe 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 skewsymmetric representation of the cross product:
In
classical mechanicsIn physics, classical mechanics is one of the two major subfields 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 crossproduct with the LeviCivita symbol can cause mechanicalsymmetries to be obvious when physicalsystems are isotropic in space. (Quick example: consider a particle in a Hooke's Law potential in threespace, free to oscillate in three dimensions; none of these dimensions are "special" in any sense, so symmetries lie in the crossproductrepresented angularmomentum which are made clear by the abovementioned LeviCivita 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,
x →
y →
z →
x. 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 abovementioned
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 graphicsComputer 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 midpoint) 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 geometryComputational 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 "righthanded" 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
MomentIn 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 forceIn 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
torqueTorque, 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 momentumIn 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 multiview 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
bivectorIn mathematics, a bivector or 2vector 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
a∧
b as the oriented parallelogram spanned by
a and
b. The cross product is then obtained by taking the
Hodge dualIn 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 finitedimensional oriented inner product space.Dimensions and algebra:...
of the bivector
a∧
b, identifying 2vectors with vectors:
This can be thought of as the oriented multidimensional 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 twodimensional – another oriented plane element. So, only in three dimensions is the cross product of
a and
b the vector dual to the bivector
a∧
b: it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as
a∧
b has relative to the unit bivector; precisely the properties described above.
Cross product and handedness
When measurable quantities involve cross products, the
handedness of the coordinate systems used cannot be arbitrary. However, when physics laws are written as equations, it should be possible to make an arbitrary choice of the coordinate system (including handedness). To avoid problems, one should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two vectors, one must take into account that when the handedness of the coordinate system is
not fixed a priori, the result is not a (true) vector but a
pseudovectorIn 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, righthanded (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 handednessfree approach is possible using
exterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higherdimensional 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 algebraIn 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, skewsymmetry, and the Jacobi identity. Many Lie algebras exist, and their study is a major field of mathematics, called
Lie theoryLie 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
quaternionIn 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 threedimensional 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 [
a_{1},
a_{2},
a_{3}] is represented as the quaternion
a_{1}i +
a_{2}j +
a_{3}k, 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 productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength 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
commutatorIn 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 7dimensional 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 algebraIn 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 2vector 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 2vectors with vectors.
The wedge product and dot product can be combined to form the
Clifford productIn mathematics, Clifford algebras are a type of associative algebra. As Kalgebras, 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 algebraIn 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 pvectors and multivectors with Grassmann algebra.Origin:In a vector space...
, the cross product can be seen as the (1,2)tensor (a
mixed tensorIn 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 3dimensional
volume formIn mathematics, a volume form on a differentiable manifold is a nowherevanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an nform, 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 nary multilinear skewsymmetric form. In the presence of a basis, such as on this is given by the 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 GstructuresIn differential geometry, a Gstructure on an nmanifold M, for a given structure group G, is a Gsubbundle of the tangent frame bundle FM of M....
, a volume form is an 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 indexIn 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 3dimensional volume form defines a product
by taking the determinant of the matrix given by these 3 vectors.
By
dualityIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional 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 productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
; more generally, a nondegenerate 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
ndimensional 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 vectorvalued product, or
 a tensor, which takes as input 2 vectors and gives as output skewsymmetric 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 skewsymmetric, and can be defined in terms of the determinant and
parityIn 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
R^{n} by:
This formula is identical in structure to the determinant formula for the normal cross product in
R^{3} except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (
v_{1},...,
v_{n1},Λ(
v_{1},...,
v_{n1})) have a positive
orientationIn 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 lefthanded or righthanded. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
with respect to (
e_{1},...,
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
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 LagrangeJosephLouis 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
tetrahedronIn 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 HamiltonSir 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
quaternionIn 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 threedimensional 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
R^{3}, their quaternion product can be summarized as [−
u·
v,
u×
v].
James Clerk MaxwellJames 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 equationsMaxwell'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 CliffordWilliam 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 DynamicElements 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
areaArea is a quantity that expresses the extent of a twodimensional 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
parallelogramIn 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 HeavisideOliver Heaviside was a selftaught 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 GibbsJosiah 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 UniversityYale 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 thirdoldest institution of higher education in the United States...
in
ConnecticutConnecticut 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 productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength 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 GrassmannHermann 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 CliffordWilliam 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 algebraIn mathematics, Clifford algebras are a type of associative algebra. As Kalgebras, 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 threedimensional 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 KotelnikovAleksandr 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 WilsonEdwin 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 productIn mathematics, the triple product is a product of three vectors. The name "triple product" is used for two different products, the scalarvalued scalar triple product and, less often, the vectorvalued 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
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
In mathematics, a bivector or 2vector 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
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