In

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.

_{1}i + ''s''_{2}j + ''s''_{3}k = are
:$\backslash begin\; s\_1\; \&=\; a\_2b\_3-a\_3b\_2\backslash \backslash \; s\_2\; \&=\; a\_3b\_1-a\_1b\_3\backslash \backslash \; s\_3\; \&=\; a\_1b\_2-a\_2b\_1\; \backslash end$
Using

^{3}

^{2} is the

^{3}, 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
:$(\backslash mathbf\; \backslash times\; \backslash mathbf)\; \backslash cdot\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; \backslash equiv\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)(\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; -\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)(\backslash mathbf\; \backslash cdot\; \backslash mathbf).$
If and this simplifies to the formula above.

^{3}. Specifically, if n is a unit vector in R^{3} 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
:$\backslash left.\; \backslash \_\; R(\backslash phi,\backslash boldsymbol)\; \backslash boldsymbol\; =\; \backslash boldsymbol\; \backslash times\; \backslash boldsymbol$
for every vector x in R^{3}. The cross product with n therefore describes the infinitesimal generator of the rotations about n. These infinitesimal generators form the ^{3} with cross product is isomorphic to the Lie algebra so(3).

^{3} 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 R^{3} and so(3). Under this map, the cross product of 3-vectors corresponds to the commutator of 3x3 skew-symmetric matrices.
:

_{ijk}'' and a dot product ''η^{mi}'', which are useful in converting vector notation for tensor applications:
:$\backslash mathbf\; =\; \backslash mathbf\; \backslash Leftrightarrow\backslash \; c^m\; =\; \backslash sum\_^3\; \backslash sum\_^3\; \backslash sum\_^3\; \backslash eta^\; E\_\; a^j\; b^k$
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
:$\backslash mathbf\; =\; \backslash mathbf\; \backslash Leftrightarrow\backslash \; c^m\; =\; \backslash eta^\; E\_\; a^j\; b^k$
in which repeated indices are summed over the values 1 to 3.
In a positively-oriented orthonormal basis ''η^{mi}'' = δ^{''mi''} (the Kronecker delta) and $E\_\; =\; \backslash varepsilon\_$ (the Levi-Civita symbol). In that case, this representation is another form of the skew-symmetric representation of the cross product:
:$;\; href="/html/ALL/s/varepsilon\_\_a^j.html"\; ;"title="varepsilon\_\; a^j">varepsilon\_\; a^j$
In

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

^{''n''} by:
:$\backslash bigwedge\_^\backslash mathbf\_i\; =\; \backslash begin\; v\_1^1\; \&\backslash cdots\; \&v\_1^\backslash \backslash \; \backslash vdots\; \&\backslash ddots\; \&\backslash vdots\backslash \backslash \; v\_^1\; \&\; \backslash cdots\; \&v\_^\backslash \backslash \; \backslash mathbf\_1\; \&\backslash cdots\; \&\backslash mathbf\_\; \backslash end.$
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_{''n''−1}, Λv_{''i''}) have a positive orientation 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 $(n-1)$-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 $;\; href="/html/ALL/s/\_1,\backslash ldots,v\_n.html"\; ;"title="\_1,\backslash ldots,v\_n">\_1,\backslash ldots,v\_n$ satisfies the Filippov identity,
:$;\; href="/html/ALL/s/x\_1,\backslash ldots,x\_n.html"\; ;"title="x\_1,\backslash ldots,x\_n">x\_1,\backslash ldots,x\_n$
and so it endows R^{n+1} with a structure of n-Lie algebra (see Proposition 1 of ).

^{3}, their quaternion product can be summarized as . 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:

A quick geometrical derivation and interpretation of cross products

created at

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

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 $\backslash times$. 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 . Inphysics
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
:$\backslash mathbf\; \backslash times\; \backslash mathbf\; =\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; \backslash sin\; (\backslash theta)\; \backslash \; \backslash mathbf$
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 :$\backslash begin\; \backslash mathbf\&\backslash times\backslash mathbf\; \&\&=\; \backslash mathbf\backslash \backslash \; \backslash mathbf\&\backslash times\backslash mathbf\; \&\&=\; \backslash mathbf\backslash \backslash \; \backslash mathbf\&\backslash times\backslash mathbf\; \&\&=\; \backslash mathbf\; \backslash end$ which imply, by the anticommutativity of the cross product, that :$\backslash begin\; \backslash mathbf\&\backslash times\backslash mathbf\; \&\&=\; -\backslash mathbf\backslash \backslash \; \backslash mathbf\&\backslash times\backslash mathbf\; \&\&=\; -\backslash mathbf\backslash \backslash \; \backslash mathbf\&\backslash times\backslash mathbf\; \&\&=\; -\backslash mathbf\; \backslash end$ The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that :$\backslash mathbf\backslash times\backslash mathbf\; =\; \backslash mathbf\backslash times\backslash mathbf\; =\; \backslash mathbf\backslash times\backslash mathbf\; =\; \backslash mathbf$ (the zero vector). These equalities, together with thedistributivity
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:
:$\backslash begin\; \backslash mathbf\; \&=\; a\_1\backslash mathbf\; \&\&+\; a\_2\backslash mathbf\; \&\&+\; a\_3\backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \&=\; b\_1\backslash mathbf\; \&\&+\; b\_2\backslash mathbf\; \&\&+\; b\_3\backslash mathbf\; \backslash end$
Their cross product can be expanded using distributivity:
:$\backslash begin\; \backslash mathbf\backslash times\backslash mathbf\; =\; \&(a\_1\backslash mathbf\; +\; a\_2\backslash mathbf\; +\; a\_3\backslash mathbf)\; \backslash times\; (b\_1\backslash mathbf\; +\; b\_2\backslash mathbf\; +\; b\_3\backslash mathbf)\backslash \backslash \; =\; \&a\_1b\_1(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; a\_1b\_2(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; a\_1b\_3(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; \backslash \backslash \; \&a\_2b\_1(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; a\_2b\_2(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; a\_2b\_3(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; \backslash \backslash \; \&a\_3b\_1(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; a\_3b\_2(\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; a\_3b\_3(\backslash mathbf\; \backslash times\; \backslash mathbf)\backslash \backslash \; \backslash end$
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:
:$\backslash begin\; \backslash mathbf\backslash times\backslash mathbf\; =\; \&\backslash quad\backslash \; a\_1b\_1\backslash mathbf\; +\; a\_1b\_2\backslash mathbf\; -\; a\_1b\_3\backslash mathbf\; \backslash \backslash \; \&-\; a\_2b\_1\backslash mathbf\; +\; a\_2b\_2\backslash mathbf\; +\; a\_2b\_3\backslash mathbf\; \backslash \backslash \; \&+\; a\_3b\_1\backslash mathbf\backslash \; -\; a\_3b\_2\backslash mathbf\backslash \; +\; a\_3b\_3\backslash mathbf\; \backslash \backslash \; =\; \&(a\_2b\_3\; -\; a\_3b\_2)\backslash mathbf\; +\; (a\_3b\_1\; -\; a\_1b\_3)\backslash mathbf\; +\; (a\_1b\_2\; -\; a\_2b\_1)\backslash mathbf\backslash \backslash \; \backslash end$
meaning that the three scalar components of the resulting vector s = ''s''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:
:$\backslash begins\_1\backslash \backslash s\_2\backslash \backslash s\_3\backslash end=\backslash begina\_2b\_3-a\_3b\_2\backslash \backslash a\_3b\_1-a\_1b\_3\backslash \backslash a\_1b\_2-a\_2b\_1\backslash end$
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. :$\backslash mathbf\; =\; \backslash begin\; \backslash mathbf\&\backslash mathbf\&\backslash mathbf\backslash \backslash \; a\_1\&a\_2\&a\_3\backslash \backslash \; b\_1\&b\_2\&b\_3\backslash \backslash \; \backslash end$ This determinant can be computed using Sarrus's rule orcofactor 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
:$\backslash begin\; \backslash mathbf\; \&=(a\_2b\_3\backslash mathbf+a\_3b\_1\backslash mathbf+a\_1b\_2\backslash mathbf)\; -\; (a\_3b\_2\backslash mathbf+a\_1b\_3\backslash mathbf+a\_2b\_1\backslash mathbf)\backslash \backslash \; \&=(a\_2b\_3\; -\; a\_3b\_2)\backslash mathbf\; +(a\_3b\_1\; -\; a\_1b\_3)\backslash mathbf\; +(a\_1b\_2\; -\; a\_2b\_1)\backslash mathbf.\; \backslash end$
Using cofactor expansion along the first row instead, it expands to
:$\backslash begin\; \backslash mathbf\; \&=\; \backslash begin\; a\_2\&a\_3\backslash \backslash \; b\_2\&b\_3\; \backslash end\backslash mathbf\; -\; \backslash begin\; a\_1\&a\_3\backslash \backslash \; b\_1\&b\_3\; \backslash end\backslash mathbf\; +\; \backslash begin\; a\_1\&a\_2\backslash \backslash \; b\_1\&b\_2\; \backslash end\backslash mathbf\; \backslash \backslash \; \&=(a\_2b\_3\; -\; a\_3b\_2)\backslash mathbf\; -(a\_1b\_3\; -\; a\_3b\_1)\backslash mathbf\; +(a\_1b\_2\; -\; a\_2b\_1)\backslash mathbf,\; \backslash end$
which gives the components of the resulting vector directly.
Using Levi-Civita tensors

* In any basis, the cross-product $a\; \backslash times\; b$ 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, $a\; \backslash times\; b$ is given by the pseudo-tensorial formula $\backslash varepsilon\_a^ib^j$ where $\backslash varepsilon\_$ 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, $a\; \backslash times\; b$ is given by the pseudo-tensorial formula $(-1)^B\backslash varepsilon\_a^ib^j$ where $(-1)^B\; =\; \backslash pm\; 1$ 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 positivearea
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):
$$\backslash left\backslash ,\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash right\backslash ,\; =\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; \backslash left,\; \backslash sin\; \backslash theta\; \backslash \; .$$
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):
:$\backslash mathbf\backslash cdot(\backslash mathbf\backslash times\; \backslash mathbf)=\; \backslash mathbf\backslash cdot(\backslash mathbf\backslash times\; \backslash mathbf)=\; \backslash mathbf\backslash cdot(\backslash mathbf\backslash times\; \backslash mathbf).$
Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value:
:$V\; =\; ,\; \backslash mathbf\; \backslash cdot\; (\backslash mathbf\; \backslash times\; \backslash mathbf),\; .$
Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of ''perpendicularity'' in the same way that the dot product is a measure of ''parallelism''. Given two unit vectors
In mathematics, 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: :$\backslash mathbf\; \backslash times\; \backslash mathbf\; =\; \backslash mathbf.$ The cross product isanticommutative
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 ...

,
:$\backslash mathbf\; \backslash times\; \backslash mathbf\; =\; -(\backslash mathbf\; \backslash times\; \backslash mathbf),$
distributive over addition,
: $\backslash mathbf\; \backslash times\; (\backslash mathbf\; +\; \backslash mathbf)\; =\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; (\backslash mathbf\; \backslash times\; \backslash mathbf),$
and compatible with scalar multiplication so that
:$(r\backslash ,\backslash mathbf)\; \backslash times\; \backslash mathbf\; =\; \backslash mathbf\; \backslash times\; (r\backslash ,\backslash mathbf)\; =\; r\backslash ,(\backslash mathbf\; \backslash times\; \backslash mathbf).$
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:
:$\backslash mathbf\; \backslash times\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; \backslash mathbf\; \backslash times\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; +\; \backslash mathbf\; \backslash times\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash mathbf.$
Distributivity, linearity and Jacobi identity show that the Rvector 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:
:$\backslash begin\; \backslash mathbf\; \&=\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; -\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\backslash \backslash \; \&=\; \backslash mathbf\; \backslash times\; (\backslash mathbf\; -\; \backslash mathbf).\backslash \backslash \; \backslash end$
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:
:$\backslash mathbf\; =\; \backslash mathbf\; +\; t\backslash ,\backslash mathbf,$
for some scalar ''t''.
If, in addition to and as above, it is the case that then
:$\backslash begin\; \backslash mathbf\; \backslash times\; (\backslash mathbf\; -\; \backslash mathbf)\; \&=\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash cdot\; (\backslash mathbf\; -\; \backslash mathbf)\; \&=\; 0,\; \backslash end$
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:
:$(R\backslash mathbf)\; \backslash times\; (R\backslash mathbf)\; =\; R(\backslash mathbf\; \backslash times\; \backslash mathbf)$, 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 $\backslash det(R)=1$.
More generally, the cross product obeys the following identity under matrix transformations:
:$(M\backslash mathbf)\; \backslash times\; (M\backslash mathbf)\; =\; (\backslash det\; M)\; \backslash left(M^\backslash right)^\backslash mathrm(\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash operatorname\; M\; (\backslash mathbf\; \backslash times\; \backslash mathbf)$
where $M$ is a 3-by-3 matrix and $\backslash left(M^\backslash right)^\backslash mathrm$ 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 $\backslash operatorname$ 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 $\backslash mathbf\; \backslash times\; \backslash mathbf$, the following relation holds true:
:$M(\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash operatorname(M)(\backslash mathbf\; \backslash times\; \backslash mathbf)\; -\; \backslash mathbf\; \backslash times\; M\backslash mathbf\; +\; \backslash mathbf\; \backslash times\; M\backslash mathbf$
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:
:$\backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash in\; NS\backslash left(\backslash begin\backslash mathbf\; \backslash \backslash \; \backslash mathbf\backslash end\backslash right).$
For the sum of two cross products, the following identity holds:
:$\backslash mathbf\; \backslash times\; \backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash mathbf\; =\; (\backslash mathbf\; -\; \backslash mathbf)\; \backslash times\; (\backslash mathbf\; -\; \backslash mathbf)\; +\; \backslash mathbf\; \backslash times\; \backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash mathbf.$
Differentiation

Theproduct 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:
:$\backslash frac(\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash frac\; \backslash times\; \backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash frac\; ,$
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 :$\backslash mathbf\; \backslash cdot\; (\backslash mathbf\; \backslash times\; \backslash mathbf),$ It is the signed volume of theparallelepiped
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:
:$\backslash mathbf\; \backslash cdot\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash mathbf\; \backslash cdot\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash mathbf\; \backslash cdot\; (\backslash mathbf\; \backslash times\; \backslash mathbf),$
The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula
:$\backslash begin\; \backslash mathbf\; \backslash times\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash mathbf(\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; -\; \backslash mathbf(\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; \backslash \backslash \; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; \backslash times\; \backslash mathbf\; =\; \backslash mathbf(\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; -\; \backslash mathbf\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; \backslash end$
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
:$\backslash begin\; \backslash nabla\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; \&=\; \backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; )\; -\; (\backslash nabla\; \backslash cdot\; \backslash nabla)\; \backslash mathbf\; \backslash \backslash \; \&=\; \backslash nabla\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; )\; -\; \backslash nabla^2\; \backslash mathbf,\backslash \backslash \; \backslash end$
where ∇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:
:$\backslash begin\; (\backslash mathbf\backslash times\; \backslash mathbf)\backslash times\; (\backslash mathbf\backslash times\; \backslash mathbf)\; \&=\; (\backslash mathbf\backslash cdot(\backslash mathbf\backslash times\; \backslash mathbf))\; \backslash mathbf\; \backslash \backslash \; (\backslash mathbf\backslash times\; \backslash mathbf)\backslash cdot(\backslash mathbf\backslash times\; \backslash mathbf)\; \&=\; \backslash mathbf^\backslash mathrm\; \backslash left(\; \backslash left(\; \backslash mathbf^\backslash mathrm\; \backslash mathbf\backslash right)I\; -\; \backslash mathbf\; \backslash mathbf^\backslash mathrm\; \backslash right)\; \backslash mathbf\backslash \backslash \; \&=\; (\backslash mathbf\backslash cdot\; \backslash mathbf)(\backslash mathbf\backslash cdot\; \backslash mathbf)-(\backslash mathbf\backslash cdot\; \backslash mathbf)\; (\backslash mathbf\backslash cdot\; \backslash mathbf)\; \backslash end$
where ''I'' is the identity matrix.
Alternative formulation

The cross product and the dot product are related by: :$\backslash left\backslash ,\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash right\backslash ,\; ^2\; =\; \backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; ^2\; \backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; ^2\; -\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)^2\; .$ 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: :$\backslash mathbf\; =\; \backslash left\backslash ,\; \backslash mathbf\; a\; \backslash right\backslash ,\; \backslash left\backslash ,\; \backslash mathbf\; b\; \backslash right\backslash ,\; \backslash cos\; \backslash theta\; ,$ the above given relationship can be rewritten as follows: :$\backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; ^2\; =\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; ^2\; \backslash left\backslash ,\; \backslash mathbf\backslash right\; \backslash ,\; ^2\; \backslash left(1-\backslash cos^2\; \backslash theta\; \backslash right)\; .$ Invoking thePythagorean 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:
:$\backslash left\backslash ,\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash right\backslash ,\; =\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; \backslash left,\; \backslash sin\; \backslash theta\; \backslash \; ,$
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: :$\backslash left\backslash ,\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash right\backslash ,\; ^2\; \backslash equiv\; \backslash det\; \backslash begin\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\backslash \backslash \; \backslash end\; \backslash equiv\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; ^2\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; ^2\; -\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)^2\; .$ can be compared with another relation involving the right-hand side, namely Lagrange's identity expressed as: :$\backslash sum\_\; \backslash left(\; a\_ib\_j\; -\; a\_jb\_i\; \backslash right)^2\; \backslash equiv\; \backslash left\backslash ,\; \backslash mathbf\; a\; \backslash right\backslash ,\; ^2\; \backslash left\backslash ,\; \backslash mathbf\; b\; \backslash right\backslash ,\; ^2\; -\; (\; \backslash mathbf\; )^2\backslash \; ,$ 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: :$\backslash begin\; \&\backslash left\backslash ,\; \backslash mathbf\; \backslash times\; \backslash mathbf\backslash right\backslash ,\; ^2\; \backslash equiv\; \backslash sum\_\backslash left(a\_ib\_j\; -\; a\_jb\_i\; \backslash right)^2\; \backslash \backslash \; \backslash equiv\; \&\backslash left(a\_1b\_2\; -\; b\_1a\_2\backslash right)^2\; +\; \backslash left(a\_2b\_3\; -\; a\_3b\_2\backslash right)^2\; +\; \backslash left(a\_3b\_1\; -\; a\_1b\_3\backslash right)^2\; \backslash \; .\; \backslash end$ The same result is found directly using the components of the cross product found from: :$\backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash equiv\; \backslash det\; \backslash begin\; \backslash hat\backslash mathbf\; \&\; \backslash hat\backslash mathbf\; \&\; \backslash hat\backslash mathbf\; \backslash \backslash \; a\_1\; \&\; a\_2\; \&\; a\_3\; \backslash \backslash \; b\_1\; \&\; b\_2\; \&\; b\_3\; \backslash \backslash \; \backslash end.$ In RInfinitesimal generators of rotations

The cross product conveniently describes the infinitesimal generators ofrotation
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 RLie 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 RAlternative 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: $$\backslash begin\; \backslash mathbf\; \backslash times\; \backslash mathbf\; =;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$ where superscript refers to thetranspose
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:
$$;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$
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,
$$;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$
or
$$;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$
where $\backslash otimes$ 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:
$$\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf$$
then
$$;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$
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
$$;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$ and $$\backslash mathbf^\backslash mathrm\; T\; \backslash ,;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$
and from fact that ''asub>× is skew-symmetric it follows that
$$\backslash mathbf^\backslash mathrm\; T\; \backslash ,;\; href="/html/ALL/s/mathbf.html"\; ;"title="mathbf">mathbf$$
The above-mentioned triple product expansion (bac–cab rule) can be easily proven using this notation.
As mentioned above, the Lie algebra RIndex notation for tensors

The cross product can alternatively be defined in terms of the Levi-Civita tensor ''Eclassical 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 :$\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf$ where: :$\backslash mathbf\; =\; \backslash begina\_x\backslash \backslash a\_y\backslash \backslash a\_z\backslash end,\backslash \; \backslash mathbf\; =\; \backslash beginb\_x\backslash \backslash b\_y\backslash \backslash b\_z\backslash end,\backslash \; \backslash mathbf\; =\; \backslash beginc\_x\backslash \backslash c\_y\backslash \backslash c\_z\backslash end$ 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 :$\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf$ then: :$\backslash mathbf\; =\; \backslash beginb\_x\backslash \backslash b\_y\backslash \backslash b\_z\backslash end\; \backslash times\; \backslash beginc\_x\backslash \backslash c\_y\backslash \backslash c\_z\backslash end.$ 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: :$a\_x\; =\; \backslash beginb\_y\backslash \backslash b\_z\backslash end\; \backslash times\; \backslash beginc\_y\backslash \backslash c\_z\backslash end.$ 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. :$a\_y\; =\; \backslash beginb\_z\backslash \backslash b\_x\backslash end\; \backslash times\; \backslash beginc\_z\backslash \backslash c\_x\backslash end,\backslash \; a\_z\; =\; \backslash beginb\_x\backslash \backslash b\_y\backslash end\; \backslash times\; \backslash beginc\_x\backslash \backslash c\_y\backslash end$ 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 twoskew 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 $p\_1=(x\_1,y\_1),\; p\_2=(x\_2,y\_2)$ and $p\_3=(x\_3,y\_3)$. It corresponds to the direction (upward or downward) of the cross product of the two coplanar vectors defined by the two pairs of points $(p\_1,\; p\_2)$ and $(p\_1,\; p\_3)$. The sign of the acute angle is the sign of the expression
:$P\; =\; (x\_2-x\_1)(y\_3-y\_1)-(y\_2-y\_1)(x\_3-x\_1),$
which is the signed length of the cross product of the two vectors.
In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around $p\_1$ from $p\_2$ to $p\_3$, otherwise a negative angle. From another point of view, the sign of $P$ tells whether $p\_3$ lies to the left or to the right of line $p\_1,\; p\_2.$
The cross product is used in calculating the volume of a polyhedron
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

Theangular 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:
: $\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf,$
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:
: $\backslash mathbf\_\backslash mathrm\; =\; \backslash mathbf\_\backslash mathrm\; \backslash times\; \backslash mathbf\_\backslash mathrm\backslash ,$
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 $\backslash mathbf$
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 arigid 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:
: $\backslash mathbf\_P\; -\; \backslash mathbf\_Q\; =\; \backslash boldsymbol\backslash omega\; \backslash times\; \backslash left(\; \backslash mathbf\_P\; -\; \backslash mathbf\_Q\; \backslash right)\backslash ,$
where $\backslash mathbf$ is the point's position, $\backslash mathbf$ is its velocity and $\backslash boldsymbol\backslash omega$ 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 $\backslash mathbf$ and velocity $\backslash mathbf$ are ''true'' vectors, the angular velocity $\backslash boldsymbol\backslash omega$ is a ''pseudovector'' or ''axial vector''.
Lorentz force

The cross product is used to describe theLorentz 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
: $\backslash mathbf\; =\; q\_e\; \backslash left(\; \backslash mathbf+\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash right)$
Since velocity force and electric field are all ''true'' vectors, the magnetic field is a ''pseudovector''.
Other

Invector 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. Inexterior 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:
: $a\; \backslash times\; b\; =\; \backslash star\; (a\; \backslash wedge\; b).$
This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented one-dimensional element – a vector – whereas, for example, in four dimensions the Hodge dual of a bivector is two-dimensional – a bivector. So, only in three dimensions can a vector cross product of ''a'' and ''b'' be defined as the vector dual to the bivector : it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as has relative to the unit bivector; precisely the properties described above.
Handedness

Consistency

When physics laws are written as equations, it is possible to make an arbitrary choice of the coordinate system, including handedness. One should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two polar vectors, one must take into account that the result is an 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 byLie 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 $\backslash mathbf^3.$ In the basis $\backslash ,$ the product is $;\; href="/html/ALL/s/,y.html"\; ;"title=",y">,y$
Quaternions

The cross product can also be described in terms ofquaternion
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 theoctonion
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-dimensionalvector 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 $\backslash mathbf\; =\; \backslash mathbf\; \backslash mathbf$, $\backslash mathbf\; =\; \backslash mathbf\; \backslash mathbf$, and $\backslash mathbf\; =\; \backslash mathbf\; \backslash mathbf$, the cross product corresponds exactly to the commutator product in geometric algebra and both use the same symbol $\backslash times$. The commutator product is defined for 2-vectors $A$ and $B$ in geometric algebra as:
: $A\; \backslash times\; B\; =\; \backslash tfrac(AB\; -\; BA)$
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: $V\backslash times\; \backslash cdots\; \backslash times\; V\; \backslash to\; \backslash mathbf.$ This is an ''n''-ary multilinear skew-symmetric form. In the presence of a basis, such as on $\backslash mathbf^n,$ 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 $V\; \backslash times\; V\; \backslash times\; V\; \backslash to\; \backslash mathbf,$ by taking the determinant of the matrix given by these 3 vectors. By duality, this is equivalent to a function $V\; \backslash times\; V\; \backslash to\; V^*,$ (fixing any two inputs gives a function $V\; \backslash to\; \backslash mathbf$ by evaluating on the third input) and in the presence of aninner 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 $V\; \backslash to\; V^*,$ and thus this yields a map $V\; \backslash times\; V\; \backslash to\; V,$ which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index".
Translating the above algebra into geometry, the function "volume of the parallelepiped defined by $(a,b,-)$" (where the first two vectors are fixed and the last is an input), which defines a function $V\; \backslash to\; \backslash mathbf$, can be ''represented'' uniquely as the dot product with a vector: this vector is the cross product $a\; \backslash times\; b.$ From this perspective, the cross product is ''defined'' by the scalar triple product, $\backslash mathrm(a,b,c)\; =\; (a\backslash times\; b)\backslash cdot\; c.$
In the same way, in higher dimensions one may define generalized cross products by raising indices of the ''n''-dimensional volume form, which is a $(0,n)$-tensor.
The most direct generalizations of the cross product are to define either:
* a $(1,n-1)$-tensor, which takes as input $n-1$ vectors, and gives as output 1 vector – an $(n-1)$-ary vector-valued product, or
* a $(n-2,2)$-tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank – a binary product with rank tensor values. One can also define $(k,n-k)$-tensors for other ''k''.
These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity.
The $(n-1)$-ary product can be described as follows: given $n-1$ vectors $v\_1,\backslash dots,v\_$ in $\backslash mathbf^n,$ define their generalized cross product $v\_n\; =\; v\_1\; \backslash times\; \backslash cdots\; \backslash times\; v\_$ 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 $v\_1,\backslash dots,v\_n$ is positively oriented.
This is the unique multilinear, alternating product which evaluates to $e\_1\; \backslash times\; \backslash cdots\; \backslash times\; e\_\; =\; e\_n$, $e\_2\; \backslash times\; \backslash cdots\; \backslash times\; e\_n\; =\; e\_1,$ and so forth for cyclic permutations of indices.
In coordinates, one can give a formula for this $(n-1)$-ary analogue of the cross product in RHistory

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 $\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf$, $\backslash boldsymbol\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf$, and $\backslash boldsymbol\; =\; \backslash mathbf\; \backslash times\; \backslash boldsymbol$. Thereby, the modern $\backslash mathbf$ corresponds to the three variables $(x,\; x\text{'},\; x\text{'}\text{'})$ 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 RJames 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 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 $u\; \backslash times\; v$ 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 $\backslash times$ 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 ...

.
See also

*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)
Notes

References

Bibliography

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

*A quick geometrical derivation and interpretation of cross products

created at

Syracuse University
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