TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
on two vectors in a three-dimensional
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
Euclidean vector space (named here $E$), and is denoted by the symbol $\times$. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to both and , and thus normal to the plane containing them. It has many applications in mathematics,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, and
computer programming Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
. It should not be confused with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
(projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are ''not'' linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) 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 eq ...
with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is
anticommutative In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
(that is, ) and is distributive over addition (that is, ). The space $E$ together with the cross product is an algebra over the real numbers, which is neither
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
nor
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, but is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
with the cross product being the Lie bracket. Like the dot product, it depends on the metric of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, but unlike the dot product, it also depends on a choice of orientation (or " handedness") of the space (it's why an oriented space is needed). In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector c ...
or 2-form result) and is independent of the orientation of the space. The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in dimensions, take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. The cross-product in seven dimensions has undesirable properties (e.g. it fails to satisfy the Jacobi identity), however, so it is not used in mathematical physics to represent quantities such as multi-dimensional
space-time In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. (See § Generalizations, below, for other dimensions.) # Definition The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by . In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...
and
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, the wedge notation is often used (in conjunction with the name ''vector product''), although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to dimensions. The cross product is defined as a vector c that is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
(orthogonal) to both a and b, with a direction given by the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
and a magnitude equal to the area of the
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) 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 eq ...
that the vectors span. The cross product is defined by the formula :$\mathbf \times \mathbf = \left\, \mathbf \right\, \left\, \mathbf \right\, \sin \left(\theta\right) \ \mathbf$ where: * ''θ'' is the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
between a and b in the plane containing them (hence, it is between 0° and 180°) * ‖a‖ and ‖b‖ are the magnitudes of vectors a and b * and n is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to the plane containing a and b, in the direction given by the right-hand rule (illustrated). If the vectors a and b are parallel (that is, the angle ''θ'' between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

## Direction By convention, the direction of the vector n is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is anti-commutative; that is, . By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector. As the cross product operator depends on the orientation of the space (as explicit in the definition above), the cross product of two vectors is not a "true" vector, but a ''pseudovector''. See for more detail.

# Names and origin In 1842,
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ire ...
discovered the algebra of
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero scalar part) is performed, it results in a quaternion with a scalar and vector part. The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors. In 1881,
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
, and independently
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ve ...
, introduced the notation for both the dot product and the cross product using a period () and an "×" (), respectively, to denote them.''A History of Vector Analysis''
by Michael J. Crowe, Math. UC Davis.
In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford coined the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature. Both the cross notation () and the name cross product were possibly inspired by the fact that each scalar component of is computed by multiplying non-corresponding components of a and b. Conversely, a dot product involves multiplications between corresponding components of a and b. As explained below, the cross product can be expressed in the form of a determinant of a special matrix. According to Sarrus's rule, this involves multiplications between matrix elements identified by crossed diagonals.

# Computing

## Coordinate notation If (i, j,k) is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities :$\begin \mathbf&\times\mathbf &&= \mathbf\\ \mathbf&\times\mathbf &&= \mathbf\\ \mathbf&\times\mathbf &&= \mathbf \end$ which imply, by the anticommutativity of the cross product, that :$\begin \mathbf&\times\mathbf &&= -\mathbf\\ \mathbf&\times\mathbf &&= -\mathbf\\ \mathbf&\times\mathbf &&= -\mathbf \end$ The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that :$\mathbf\times\mathbf = \mathbf\times\mathbf = \mathbf\times\mathbf = \mathbf$ (the zero vector). These equalities, together with the
distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic ...
and
linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
of the cross product (though neither follows easily from the definition given above), are sufficient to determine the cross product of any two vectors a and b. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors: :$\begin \mathbf &= a_1\mathbf &&+ a_2\mathbf &&+ a_3\mathbf \\ \mathbf &= b_1\mathbf &&+ b_2\mathbf &&+ b_3\mathbf \end$ Their cross product can be expanded using distributivity: :$\begin \mathbf\times\mathbf = &\left(a_1\mathbf + a_2\mathbf + a_3\mathbf\right) \times \left(b_1\mathbf + b_2\mathbf + b_3\mathbf\right)\\ = &a_1b_1\left(\mathbf \times \mathbf\right) + a_1b_2\left(\mathbf \times \mathbf\right) + a_1b_3\left(\mathbf \times \mathbf\right) + \\ &a_2b_1\left(\mathbf \times \mathbf\right) + a_2b_2\left(\mathbf \times \mathbf\right) + a_2b_3\left(\mathbf \times \mathbf\right) + \\ &a_3b_1\left(\mathbf \times \mathbf\right) + a_3b_2\left(\mathbf \times \mathbf\right) + a_3b_3\left(\mathbf \times \mathbf\right)\\ \end$ This can be interpreted as the decomposition of into the sum of nine simpler cross products involving vectors aligned with i, j, or k. Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using the above-mentioned equalities and collecting similar terms, we obtain: :$\begin \mathbf\times\mathbf = &\quad\ a_1b_1\mathbf + a_1b_2\mathbf - a_1b_3\mathbf \\ &- a_2b_1\mathbf + a_2b_2\mathbf + a_2b_3\mathbf \\ &+ a_3b_1\mathbf\ - a_3b_2\mathbf\ + a_3b_3\mathbf \\ = &\left(a_2b_3 - a_3b_2\right)\mathbf + \left(a_3b_1 - a_1b_3\right)\mathbf + \left(a_1b_2 - a_2b_1\right)\mathbf\\ \end$ meaning that the three scalar components of the resulting vector s = ''s''1i + ''s''2j + ''s''3k = are :$\begin s_1 &= a_2b_3-a_3b_2\\ s_2 &= a_3b_1-a_1b_3\\ s_3 &= a_1b_2-a_2b_1 \end$ Using
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, ...
s, we can represent the same result as follows: :$\begins_1\\s_2\\s_3\end=\begina_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end$

## Matrix notation The cross product can also be expressed as the formal determinant:Here, "formal" means that this notation has the form of a determinant, but does not strictly adhere to the definition; it is a mnemonic used to remember the expansion of the cross product. :$\mathbf = \begin \mathbf&\mathbf&\mathbf\\ a_1&a_2&a_3\\ b_1&b_2&b_3\\ \end$ This determinant can be computed using Sarrus's rule or
cofactor expansion In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . Sp ...
. Using Sarrus's rule, it expands to :$\begin \mathbf &=\left(a_2b_3\mathbf+a_3b_1\mathbf+a_1b_2\mathbf\right) - \left(a_3b_2\mathbf+a_1b_3\mathbf+a_2b_1\mathbf\right)\\ &=\left(a_2b_3 - a_3b_2\right)\mathbf +\left(a_3b_1 - a_1b_3\right)\mathbf +\left(a_1b_2 - a_2b_1\right)\mathbf. \end$ Using cofactor expansion along the first row instead, it expands to :$\begin \mathbf &= \begin a_2&a_3\\ b_2&b_3 \end\mathbf - \begin a_1&a_3\\ b_1&b_3 \end\mathbf + \begin a_1&a_2\\ b_1&b_2 \end\mathbf \\ &=\left(a_2b_3 - a_3b_2\right)\mathbf -\left(a_1b_3 - a_3b_1\right)\mathbf +\left(a_1b_2 - a_2b_1\right)\mathbf, \end$ which gives the components of the resulting vector directly.

## Using Levi-Civita tensors

* In any basis, the cross-product $a \times b$ is given by the tensorial formula $E_a^ib^j$ where $E_$ is the covariant Levi-Civita tensor (we note the position of the indices). That corresponds to the intrinsic formula given here. * In an orthonormal basis having the same orientation as the space, $a \times b$ is given by the pseudo-tensorial formula $\varepsilon_a^ib^j$ where $\varepsilon_$ is the Levi-Civita symbol (which is a pseudo-tensor). That’s the formula used for everyday physics but it works only for this special choice of basis. * In any orthonormal basis, $a \times b$ is given by the pseudo-tensorial formula $\left(-1\right)^B\varepsilon_a^ib^j$ where $\left(-1\right)^B = \pm 1$ indicates whether the basis has the same orientation as the space or not. The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis.

# Properties

## Geometric meaning  The magnitude of the cross product can be interpreted as the positive
area Area is the quantity that expresses the extent of a Region (mathematics), region on the plane (geometry), plane or on a curved surface (mathematics), surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar ...
of the
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) 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 eq ...
having a and b as sides (see Figure 1): $\left\, \mathbf \times \mathbf \right\, = \left\, \mathbf \right\, \left\, \mathbf \right\, \left, \sin \theta \ .$ Indeed, one can also compute the volume ''V'' of a
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
having a, b and c as edges by using a combination of a cross product and a dot product, called scalar triple product (see Figure 2): :$\mathbf\cdot\left(\mathbf\times \mathbf\right)= \mathbf\cdot\left(\mathbf\times \mathbf\right)= \mathbf\cdot\left(\mathbf\times \mathbf\right).$ Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value: :$V = , \mathbf \cdot \left(\mathbf \times \mathbf\right), .$ Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of ''perpendicularity'' in the same way that the dot product is a measure of ''parallelism''. Given two
unit vectors In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction ve ...
, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel. Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive).

## Algebraic properties  If the cross product of two vectors is the zero vector (that is, ), then either one or both of the inputs is the zero vector, ( or ) or else they are parallel or antiparallel () so that the sine of the angle between them is zero ( or and ). The self cross product of a vector is the zero vector: :$\mathbf \times \mathbf = \mathbf.$ The cross product is
anticommutative In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
, :$\mathbf \times \mathbf = -\left(\mathbf \times \mathbf\right),$ distributive over addition, : $\mathbf \times \left(\mathbf + \mathbf\right) = \left(\mathbf \times \mathbf\right) + \left(\mathbf \times \mathbf\right),$ and compatible with scalar multiplication so that :$\left(r\,\mathbf\right) \times \mathbf = \mathbf \times \left(r\,\mathbf\right) = r\,\left(\mathbf \times \mathbf\right).$ It is not
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, but satisfies the Jacobi identity: :$\mathbf \times \left(\mathbf \times \mathbf\right) + \mathbf \times \left(\mathbf \times \mathbf\right) + \mathbf \times \left(\mathbf \times \mathbf\right) = \mathbf.$ Distributivity, linearity and Jacobi identity show that the R3
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
together with vector addition and the cross product forms a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
, the Lie algebra of the real
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
in 3 dimensions,
SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a ...
. The cross product does not obey the cancellation law; that is, with does not imply , but only that: :$\begin \mathbf &= \left(\mathbf \times \mathbf\right) - \left(\mathbf \times \mathbf\right)\\ &= \mathbf \times \left(\mathbf - \mathbf\right).\\ \end$ This can be the case where b and c cancel, but additionally where a and are parallel; that is, they are related by a scale factor ''t'', leading to: :$\mathbf = \mathbf + t\,\mathbf,$ for some scalar ''t''. If, in addition to and as above, it is the case that then :$\begin \mathbf \times \left(\mathbf - \mathbf\right) &= \mathbf \\ \mathbf \cdot \left(\mathbf - \mathbf\right) &= 0, \end$ As cannot be simultaneously parallel (for the cross product to be 0) and perpendicular (for the dot product to be 0) to a, it must be the case that b and c cancel: . From the geometrical definition, the cross product is invariant under proper rotations about the axis defined by . In formulae: :$\left(R\mathbf\right) \times \left(R\mathbf\right) = R\left(\mathbf \times \mathbf\right)$, where $R$ is a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end ...
with $\det\left(R\right)=1$. More generally, the cross product obeys the following identity under matrix transformations: :$\left(M\mathbf\right) \times \left(M\mathbf\right) = \left(\det M\right) \left\left(M^\right\right)^\mathrm\left(\mathbf \times \mathbf\right) = \operatorname M \left(\mathbf \times \mathbf\right)$ where $M$ is a 3-by-3 matrix and $\left\left(M^\right\right)^\mathrm$ is the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
of the inverse and $\operatorname$ is the cofactor matrix. It can be readily seen how this formula reduces to the former one if $M$ is a rotation matrix. If $M$ is a 3-by-3 symmetric matrix applied to a generic cross product $\mathbf \times \mathbf$, the following relation holds true: :$M\left(\mathbf \times \mathbf\right) = \operatorname\left(M\right)\left(\mathbf \times \mathbf\right) - \mathbf \times M\mathbf + \mathbf \times M\mathbf$ The cross product of two vectors lies in the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel o ...
of the matrix with the vectors as rows: :$\mathbf \times \mathbf \in NS\left\left(\begin\mathbf \\ \mathbf\end\right\right).$ For the sum of two cross products, the following identity holds: :$\mathbf \times \mathbf + \mathbf \times \mathbf = \left(\mathbf - \mathbf\right) \times \left(\mathbf - \mathbf\right) + \mathbf \times \mathbf + \mathbf \times \mathbf.$

## Differentiation

The
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
of differential calculus applies to any bilinear operation, and therefore also to the cross product: :$\frac\left(\mathbf \times \mathbf\right) = \frac \times \mathbf + \mathbf \times \frac ,$ where a and b are vectors that depend on the real variable ''t''.

## Triple product expansion

The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as :$\mathbf \cdot \left(\mathbf \times \mathbf\right),$ It is the signed volume of the
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
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: :$\mathbf \cdot \left(\mathbf \times \mathbf\right) = \mathbf \cdot \left(\mathbf \times \mathbf\right) = \mathbf \cdot \left(\mathbf \times \mathbf\right),$ 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 :$\begin \mathbf \times \left(\mathbf \times \mathbf\right) = \mathbf\left(\mathbf \cdot \mathbf\right) - \mathbf\left(\mathbf \cdot \mathbf\right) \\ \left(\mathbf \times \mathbf\right) \times \mathbf = \mathbf\left(\mathbf \cdot \mathbf\right) - \mathbf \left(\mathbf \cdot \mathbf\right) \end$ The
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and image ...
"BAC minus CAB" is used to remember the order of the vectors in the right hand member. This formula is used in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...
to simplify vector calculations. A special case, regarding
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the grad ...
s and useful in
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
, is :$\begin \nabla \times \left(\nabla \times \mathbf\right) &= \nabla \left(\nabla \cdot \mathbf \right) - \left(\nabla \cdot \nabla\right) \mathbf \\ &= \nabla \left(\nabla \cdot \mathbf \right) - \nabla^2 \mathbf,\\ \end$ where ∇2 is the
vector Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
operator. Other identities relate the cross product to the scalar triple product: :$\begin \left(\mathbf\times \mathbf\right)\times \left(\mathbf\times \mathbf\right) &= \left(\mathbf\cdot\left(\mathbf\times \mathbf\right)\right) \mathbf \\ \left(\mathbf\times \mathbf\right)\cdot\left(\mathbf\times \mathbf\right) &= \mathbf^\mathrm \left\left( \left\left( \mathbf^\mathrm \mathbf\right\right)I - \mathbf \mathbf^\mathrm \right\right) \mathbf\\ &= \left(\mathbf\cdot \mathbf\right)\left(\mathbf\cdot \mathbf\right)-\left(\mathbf\cdot \mathbf\right) \left(\mathbf\cdot \mathbf\right) \end$ where ''I'' is the identity matrix.

## Alternative formulation

The cross product and the dot product are related by: :$\left\, \mathbf \times \mathbf \right\, ^2 = \left\, \mathbf\right\, ^2 \left\, \mathbf\right\, ^2 - \left(\mathbf \cdot \mathbf\right)^2 .$ 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: :$\mathbf = \left\, \mathbf a \right\, \left\, \mathbf b \right\, \cos \theta ,$ the above given relationship can be rewritten as follows: :$\left\, \mathbf \right\, ^2 = \left\, \mathbf \right\, ^2 \left\, \mathbf\right \, ^2 \left\left(1-\cos^2 \theta \right\right) .$ Invoking the
Pythagorean trigonometric identity The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations ...
one obtains: :$\left\, \mathbf \times \mathbf \right\, = \left\, \mathbf \right\, \left\, \mathbf \right\, \left, \sin \theta \ ,$ which is the magnitude of the cross product expressed in terms of ''θ'', equal to the area of the parallelogram defined by a and b (see
definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional defini ...
above). The combination of this requirement and the property that the cross product be orthogonal to its constituents a and b provides an alternative definition of the cross product.

## Lagrange's identity

The relation: :$\left\, \mathbf \times \mathbf \right\, ^2 \equiv \det \begin \mathbf \cdot \mathbf & \mathbf \cdot \mathbf \\ \mathbf \cdot \mathbf & \mathbf \cdot \mathbf\\ \end \equiv \left\, \mathbf \right\, ^2 \left\, \mathbf \right\, ^2 - \left(\mathbf \cdot \mathbf\right)^2 .$ can be compared with another relation involving the right-hand side, namely Lagrange's identity expressed as: :$\sum_ \left\left( a_ib_j - a_jb_i \right\right)^2 \equiv \left\, \mathbf a \right\, ^2 \left\, \mathbf b \right\, ^2 - \left( \mathbf \right)^2\ ,$ where a and b may be ''n''-dimensional vectors. This also shows that the Riemannian volume form for surfaces is exactly the surface element from vector calculus. In the case where , combining these two equations results in the expression for the magnitude of the cross product in terms of its components: :$\begin &\left\, \mathbf \times \mathbf\right\, ^2 \equiv \sum_\left\left(a_ib_j - a_jb_i \right\right)^2 \\ \equiv &\left\left(a_1b_2 - b_1a_2\right\right)^2 + \left\left(a_2b_3 - a_3b_2\right\right)^2 + \left\left(a_3b_1 - a_1b_3\right\right)^2 \ . \end$ The same result is found directly using the components of the cross product found from: :$\mathbf \times \mathbf \equiv \det \begin \hat\mathbf & \hat\mathbf & \hat\mathbf \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end.$ In R3, Lagrange's equation is a special case of the multiplicativity of the norm in the quaternion algebra. It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the Binet–Cauchy identity:by :$\left(\mathbf \times \mathbf\right) \cdot \left(\mathbf \times \mathbf\right) \equiv \left(\mathbf \cdot \mathbf\right)\left(\mathbf \cdot \mathbf\right) - \left(\mathbf \cdot \mathbf\right)\left(\mathbf \cdot \mathbf\right).$ If and this simplifies to the formula above.

## Infinitesimal generators of rotations

The cross product conveniently describes the infinitesimal generators of
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s in R3. Specifically, if n is a unit vector in R3 and ''R''(''φ'', n) denotes a rotation about the axis through the origin specified by n, with angle φ (measured in radians, counterclockwise when viewed from the tip of n), then :$\left. \_ R\left(\phi,\boldsymbol\right) \boldsymbol = \boldsymbol \times \boldsymbol$ for every vector x in R3. The cross product with n therefore describes the infinitesimal generator of the rotations about n. These infinitesimal generators form the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
so(3) of the
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a t ...
, and we obtain the result that the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3).

# Alternative ways to compute

## Conversion to matrix multiplication

The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector: where superscript refers to the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
operation, and ''asub>× is defined by: The columns ''asub>×,i of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with
unit vectors In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction ve ...
. That is, or where $\otimes$ is the
outer product In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of n ...
operator. Also, if a is itself expressed as a cross product: $\mathbf = \mathbf \times \mathbf$ then This result can be generalized to higher dimensions using
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...
. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector. In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product, with the bivector instead of its vector dual. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors. This notation is also often much easier to work with, for example, in
epipolar geometry Epipolar geometry is the geometry of stereo vision. 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 ...
. From the general properties of the cross product follows immediately that   and   and from fact that ''asub>× is skew-symmetric it follows that The above-mentioned triple product expansion (bac–cab rule) can be easily proven using this notation. As mentioned above, the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3), whose elements can be identified with the 3×3 skew-symmetric matrices. The map a → ''asub>× provides an isomorphism between R3 and so(3). Under this map, the cross product of 3-vectors corresponds to the commutator of 3x3 skew-symmetric matrices. :

## Index notation for tensors

The cross product can alternatively be defined in terms of the Levi-Civita tensor ''Eijk'' and a dot product ''ηmi'', which are useful in converting vector notation for tensor applications: :$\mathbf = \mathbf \Leftrightarrow\ c^m = \sum_^3 \sum_^3 \sum_^3 \eta^ E_ a^j b^k$ where the indices $i,j,k$ correspond to vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention as :$\mathbf = \mathbf \Leftrightarrow\ c^m = \eta^ E_ a^j b^k$ in which repeated indices are summed over the values 1 to 3. In a positively-oriented orthonormal basis ''ηmi'' = δ''mi'' (the Kronecker delta) and $E_ = \varepsilon_$ (the Levi-Civita symbol). In that case, this representation is another form of the skew-symmetric representation of the cross product: : In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mech ...
: representing the cross product by using the Levi-Civita symbol can cause mechanical symmetries to be obvious when physical systems are
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
. (An example: consider a particle in a Hooke's Law potential in three-space, free to oscillate in three dimensions; none of these dimensions are "special" in any sense, so symmetries lie in the cross-product-represented angular momentum, which are made clear by the abovementioned Levi-Civita representation).

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

## Cross visualization

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

# Applications

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

## Computational geometry

The cross product appears in the calculation of the distance of two
skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the ...
(lines not in the same plane) from each other in three-dimensional space. The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle. In computational geometry of the plane, the cross product is used to determine the sign of the
acute angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
defined by three points $p_1=\left(x_1,y_1\right), p_2=\left(x_2,y_2\right)$ and $p_3=\left(x_3,y_3\right)$. It corresponds to the direction (upward or downward) of the cross product of the two coplanar vectors defined by the two pairs of points $\left(p_1, p_2\right)$ and $\left(p_1, p_3\right)$. The sign of the acute angle is the sign of the expression :$P = \left(x_2-x_1\right)\left(y_3-y_1\right)-\left(y_2-y_1\right)\left(x_3-x_1\right),$ which is the signed length of the cross product of the two vectors. In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around $p_1$ from $p_2$ to $p_3$, otherwise a negative angle. From another point of view, the sign of $P$ tells whether $p_3$ lies to the left or to the right of line $p_1, p_2.$ The cross product is used in calculating the volume of a
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
such as a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
or
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
.

## Angular momentum and torque

The
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system ...
of a particle about a given origin is defined as: : $\mathbf = \mathbf \times \mathbf,$ where is the position vector of the particle relative to the origin, is the linear momentum of the particle. In the same way, the moment of a force applied at point B around point A is given as: : $\mathbf_\mathrm = \mathbf_\mathrm \times \mathbf_\mathrm\,$ In mechanics the ''moment of a force'' is also called ''
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
'' and written as $\mathbf$ Since position linear momentum and force are all ''true'' vectors, both the angular momentum and the moment of a force are ''pseudovectors'' or ''axial vectors''.

## Rigid body

The cross product frequently appears in the description of rigid motions. Two points ''P'' and ''Q'' on a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
can be related by: : $\mathbf_P - \mathbf_Q = \boldsymbol\omega \times \left\left( \mathbf_P - \mathbf_Q \right\right)\,$ where $\mathbf$ is the point's position, $\mathbf$ is its velocity and $\boldsymbol\omega$ is the body's
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
. Since position $\mathbf$ and velocity $\mathbf$ are ''true'' vectors, the angular velocity $\boldsymbol\omega$ is a ''pseudovector'' or ''axial vector''.

## Lorentz force

The cross product is used to describe the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elec ...
experienced by a moving electric charge : $\mathbf = q_e \left\left( \mathbf+ \mathbf \times \mathbf \right\right)$ Since velocity force and electric field are all ''true'' vectors, the magnetic field is a ''pseudovector''.

## Other

In
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
, the cross product is used to define the formula for the vector operator
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
. 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.

# As an external product The cross product can be defined in terms of the exterior product. It can be generalized to an external product in other than three dimensions. This view allows a natural geometric interpretation of the cross product. In
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
the exterior product of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors ''a'' and ''b'', one can view the bivector as the oriented parallelogram spanned by ''a'' and ''b''. The cross product is then obtained by taking the Hodge star of the bivector , mapping 2-vectors to vectors: : $a \times b = \star \left(a \wedge b\right).$ This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented one-dimensional element – a vector – whereas, for example, in four dimensions the Hodge dual of a bivector is two-dimensional – a bivector. So, only in three dimensions can a vector cross product of ''a'' and ''b'' be defined as the vector dual to the bivector : it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as has relative to the unit bivector; precisely the properties described above.

# Handedness

## Consistency

When physics laws are written as equations, it is possible to make an arbitrary choice of the coordinate system, including handedness. One should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two polar vectors, one must take into account that the result is an axial vector. Therefore, for consistency, the other side must also be an axial vector. More generally, the result of a cross product may be either a polar vector or an axial vector, depending on the type of its operands (polar vectors or axial vectors). Namely, polar vectors and axial vectors are interrelated in the following ways under application of the cross product: * polar vector × polar vector = axial vector * axial vector × axial vector = axial vector * polar vector × axial vector = polar vector * axial vector × polar vector = polar vector or symbolically * polar × polar = axial * axial × axial = axial * polar × axial = polar * axial × polar = polar Because the cross product may also be a polar vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a polar vector and the other one is an axial vector (e.g., the cross product of two polar vectors). For instance, a vector triple product involving three polar vectors is a polar vector. A handedness-free approach is possible using exterior algebra.

## The paradox of the orthonormal basis

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

# Generalizations

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

## Lie algebra

The cross product can be seen as one of the simplest Lie products, and is thus generalized by
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
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 In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...
. For example, the Heisenberg algebra gives another Lie algebra structure on $\mathbf^3.$ In the basis $\,$ the product is

## Quaternions

The cross product can also be described in terms of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s. In general, if a vector is represented as the quaternion , the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors.

## Octonions

A cross product for 7-dimensional vectors can be obtained in the same way by using the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s instead of the quaternions. The nonexistence of nontrivial vector-valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem that the only
normed division algebra In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic f ...
s are the ones with dimension 1, 2, 4, and 8.

## Exterior product

In general dimension, there is no direct analogue of the binary cross product that yields specifically a vector. There is however the exterior product, which has similar properties, except that the exterior product of two vectors is now a 2-vector instead of an ordinary vector. As mentioned above, the cross product can be interpreted as the exterior product in three dimensions by using the Hodge star operator to map 2-vectors to vectors. The Hodge dual of the exterior product yields an -vector, which is a natural generalization of the cross product in any number of dimensions. The exterior product and dot product can be combined (through summation) to form the geometric product in geometric algebra.

## External product

As mentioned above, the cross product can be interpreted in three dimensions as the Hodge dual of the exterior product. In any finite ''n'' dimensions, the Hodge dual of the exterior product of vectors is a vector. So, instead of a binary operation, in arbitrary finite dimensions, the cross product is generalized as the Hodge dual of the exterior product of some given vectors. This generalization is called external product.

## Commutator product

Interpreting the three-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
of the algebra as the 2-vector (not the 1-vector) subalgebra of the three-dimensional geometric algebra, where $\mathbf = \mathbf \mathbf$, $\mathbf = \mathbf \mathbf$, and $\mathbf = \mathbf \mathbf$, the cross product corresponds exactly to the commutator product in geometric algebra and both use the same symbol $\times$. The commutator product is defined for 2-vectors $A$ and $B$ in geometric algebra as: : $A \times B = \tfrac\left(AB - BA\right)$ where $AB$ is the geometric product. The commutator product could be generalised to arbitrary multivectors in three dimensions, which results in a multivector consisting of only elements of grades 1 (1-vectors/ true vectors) and 2 (2-vectors/pseudovectors). While the commutator product of two 1-vectors is indeed the same as the exterior product and yields a 2-vector, the commutator of a 1-vector and a 2-vector yields a true vector, corresponding instead to the left and right contractions in geometric algebra. The commutator product of two 2-vectors has no corresponding equivalent product, which is why the commutator product is defined in the first place for 2-vectors. Furthermore, the commutator triple product of three 2-vectors is the same as the vector triple product of the same three pseudovectors in vector algebra. However, the commutator triple product of three 1-vectors in geometric algebra is instead the negative of the vector triple product of the same three true vectors in vector algebra. Generalizations to higher dimensions is provided by the same commutator product of 2-vectors in higher-dimensional geometric algebras, but the 2-vectors are no longer pseudovectors. Just as the commutator product/cross product of 2-vectors in three dimensions correspond to the simplest Lie algebra, the 2-vector subalgebras of higher dimensional geometric algebra equipped with the commutator product also correspond to the Lie algebras. Also as in three dimensions, the commutator product could be further generalised to arbitrary multivectors.

## Multilinear algebra

In the context of multilinear algebra, the cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form,By a volume form one means a function that takes in ''n'' vectors and gives out a scalar, the volume of the parallelotope defined by the vectors: $V\times \cdots \times V \to \mathbf.$ This is an ''n''-ary multilinear skew-symmetric form. In the presence of a basis, such as on $\mathbf^n,$ this is given by the determinant, but in an abstract vector space, this is added structure. In terms of ''G''-structures, a volume form is an $SL$-structure. a (0,3)-tensor, by raising an index. In detail, the 3-dimensional volume form defines a product $V \times V \times V \to \mathbf,$ by taking the determinant of the matrix given by these 3 vectors. By duality, this is equivalent to a function $V \times V \to V^*,$ (fixing any two inputs gives a function $V \to \mathbf$ by evaluating on the third input) and in the presence of an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
(such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism $V \to V^*,$ and thus this yields a map $V \times V \to V,$ which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index". Translating the above algebra into geometry, the function "volume of the parallelepiped defined by $\left(a,b,-\right)$" (where the first two vectors are fixed and the last is an input), which defines a function $V \to \mathbf$, can be ''represented'' uniquely as the dot product with a vector: this vector is the cross product $a \times b.$ From this perspective, the cross product is ''defined'' by the scalar triple product, $\mathrm\left(a,b,c\right) = \left(a\times b\right)\cdot c.$ In the same way, in higher dimensions one may define generalized cross products by raising indices of the ''n''-dimensional volume form, which is a $\left(0,n\right)$-tensor. The most direct generalizations of the cross product are to define either: * a $\left(1,n-1\right)$-tensor, which takes as input $n-1$ vectors, and gives as output 1 vector – an $\left(n-1\right)$-ary vector-valued product, or * a $\left(n-2,2\right)$-tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank – a binary product with rank tensor values. One can also define $\left(k,n-k\right)$-tensors for other ''k''. These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity. The $\left(n-1\right)$-ary product can be described as follows: given $n-1$ vectors $v_1,\dots,v_$ in $\mathbf^n,$ define their generalized cross product $v_n = v_1 \times \cdots \times v_$ as: * perpendicular to the hyperplane defined by the $v_i,$ * magnitude is the volume of the parallelotope defined by the $v_i,$ which can be computed as the Gram determinant of the $v_i,$ * oriented so that $v_1,\dots,v_n$ is positively oriented. This is the unique multilinear, alternating product which evaluates to $e_1 \times \cdots \times e_ = e_n$, $e_2 \times \cdots \times e_n = e_1,$ and so forth for cyclic permutations of indices. In coordinates, one can give a formula for this $\left(n-1\right)$-ary analogue of the cross product in R''n'' by: :$\bigwedge_^\mathbf_i = \begin v_1^1 &\cdots &v_1^\\ \vdots &\ddots &\vdots\\ v_^1 & \cdots &v_^\\ \mathbf_1 &\cdots &\mathbf_ \end.$ This formula is identical in structure to the determinant formula for the normal cross product in R3 except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v1, ..., v''n''−1, Λv''i'') have a positive orientation with respect to (e1, ..., e''n''). If ''n'' is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that ''n'' is even, however, the distinction must be kept. This $\left(n-1\right)$-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. Moreover, the product satisfies the Filippov identity, : and so it endows Rn+1 with a structure of n-Lie algebra (see Proposition 1 of ).

# History

In 1773, Joseph-Louis Lagrange used the component form of both the dot and cross products in order to study the tetrahedron in three dimensions.In modern notation, Lagrange defines $\mathbf = \mathbf \times \mathbf$, $\boldsymbol = \mathbf \times \mathbf$, and $\boldsymbol = \mathbf \times \boldsymbol$. Thereby, the modern $\mathbf$ corresponds to the three variables $\left(x, x\text{'}, x\text{'}\text{'}\right)$ in Lagrange's notation. In 1843,
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ire ...
introduced the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
product, and with it the terms ''vector'' and ''scalar''. Given two quaternions and , where u and v are vectors in R3, their quaternion product can be summarized as .
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education. In 1844,
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His ma ...
published a geometric algebra not tied to dimension two or three. Grassmann develops several products, including a cross product represented then by . (''See also:
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
.'') In 1853,
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
, a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product. In 1878, William Kingdon Clifford published '' Elements of Dynamic'', in which the term ''vector product'' is attested. In the book, this product of two vectors is defined to have magnitude equal to the
area Area is the quantity that expresses the extent of a Region (mathematics), region on the plane (geometry), plane or on a curved surface (mathematics), surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar ...
of the
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) 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 eq ...
of which they are two sides, and direction perpendicular to their plane. (''See also:
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomp ...
.'') In 1881 lecture notes, Gibbs represents the cross product by $u \times v$ and calls it the ''skew product''. In 1901, Gibb's student Edwin Bidwell Wilson edits and extends these lecture notes into the
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textboo ...
''
Vector Analysis Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
''. Wilson keeps the term ''skew product'', but observes that the alternative terms ''cross product''since is read as " cross " and ''vector product'' were more frequent. In 1908, Cesare Burali-Forti and Roberto Marcolongo introduce the vector product notation . This is used in
France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans. Its metropolitan area ...
and other areas until this day, as the symbol $\times$ is already used to denote
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
and the
cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
.

*
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
– a product of two sets * Geometric algebra: Rotating systems * Multiple cross products – products involving more than three vectors *
Multiplication of vectors In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: * Dot product – also known as the "scalar product", a binary operation that takes two vector ...
* Quadruple product * × (the symbol)

# Bibliography

* * * E. A. Milne (1948) Vectorial Mechanics, Chapter 2: Vector Product, pp 11 –31, London: Methuen Publishing. * *

*
A quick geometrical derivation and interpretation of cross products

created at
Syracuse University Syracuse University (informally 'Cuse or SU) is a private research university in Syracuse, New York. Established in 1870 with roots in the Methodist Episcopal Church, the university has been nonsectarian since 1920. Located in the city's Un ...
– (requires
java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
)
W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).

The vector product
Mathcentre (UK), 2009 {{DEFAULTSORT:Cross Product Bilinear maps Operations on vectors Analytic geometry