Cross Product

Information about Cross Product

For the cross product in algebraic topology, see .

For the crossed product in algebra and functional analysis, see .

For volume calculation, see .

In mathematics, the cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result. In many engineering and physics problems, it is handy to be able to construct a perpendicular vector from two existing vectors, and the cross product provides a means for doing so. The cross product is also known as the vector product, or Gibbs vector product.

The cross product is not defined except in three-dimensions (and the algebra defined by the cross product is not associative). Like the dot product, it depends on the metric of Euclidean space. Unlike the dot product, it also depends on the choice of orientation or "handedness". Certain features of the cross product can be generalized to other situations. For arbitrary choices of orientation, the cross product must be regarded not as a vector, but as a pseudovector. For arbitrary choices of metric, and in arbitrary dimensions, the cross product can be generalized by the exterior product of vectors, defining a two-form instead of a vector.

Illustration of the cross-product in respect to a right-handed coordinate system.

Definition

The cross product of two vectors a and b is denoted by a × b. In a three-dimensional Euclidean space, with a usual right-handed coordinate system, it is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

The cross product is given by the formula

$\text{[math]}$

where θ is the measure of the angle between a and b (0° ≤ θ ≤ 180°), a and b are the magnitudes of vectors a and b, and $\text{[math]}$ is a unit vector perpendicular to the plane containing a and b. If the vectors a and b are collinear (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

The direction of the vector $\text{[math]}$ 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 $\text{[math]}$ is coming out of the thumb (see the picture on the right).

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

This, however, creates a problem because transforming from one arbitrary reference system to another (e.g., a mirror image transformation from a right-handed to a left-handed coordinate system), should not change the direction of $\text{[math]}$ . The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector. See cross product and handedness for more detail.

Examples

Example 1

Consider two vectors, a = (1,2,3) and b = (4,5,6). The cross product a × b is

a × b = (1,2,3) × (4,5,6) = ((2 × 6 - 3 × 5),-(6 × 1 - 4 × 3), (1 × 5 - 2 × 4)) = (-3,6,-3).

Example 2

Consider two vectors, a = (3,0,0) and b = (0,2,0). The cross product a × b is

a × b = (3,0,0) × (0,2,0) = ((0 × 0 - 0 × 2), (0 × 0 - 3 × 0), (3 × 2 - 0 × 0)) = (0,0,6).

This example has the following interpretations:

The area of the parallelogram (a rectangle in this case) is 2 × 3 = 6.
The cross product of any two vectors in the xy plane will be parallel to the z axis.
Since the z-component of the result is positive, the non-obtuse angle from a to b is counterclockwise (when observed from a point on the +z semiaxis, and when the coordinate system is right handed).

Properties

Geometric meaning

The magnitude of the cross product can be interpreted as the unsigned area of the parallelogram having a and b as sides:

$\text{[math]}$

Algebraic properties

The cross product is anticommutative,

a × b = −b × a,

distributive over addition,

a × (b + c) = (a × b) + (a × c),

and compatible with scalar multiplication so that

(ra) × b = a × (rb) = r(a × b).

It is not associative, but satisfies the Jacobi identity:

a × (b × c) + b × (c × a) + c × (a × b) = 0.

It does not obey the cancellation law:

If a × b = a × c and a ≠ 0 then we can write:

(a × b) − (a × c) = 0 and, by the distributive law above:

a × (b − c) = 0

Now, if a is parallel to (b − c), then even if a ≠ 0 it is possible that (b − c) ≠ 0 and therefore that b ≠ c.

However, if both a · b = a · c and a × b = a × c, then we can conclude that b = c. This is because if (b − c) ≠ 0, then it obviously cannot be both parallel and perpendicular to another nonzero vector a.

The distributivity, linearity and Jacobi identity show that R³ together with vector addition and cross product forms a Lie algebra.

Further, two non-zero vectors a and b are parallel iff a × b = 0.

Lagrange's formula (triple product expansion)

Main article: Lagrange's formula

This is a very useful identity involving the cross-product. It is written as

a × (b × c) = b(a · c) − c(a · b),

which is easier to remember as “BAC minus CAB”, keeping in mind which vectors are dotted together. This formula is very useful in simplifying vector calculations in physics. A special case, regarding gradients and useful in vector calculus, is given below.

$\text{[math]}$

This is a special case of the more general Laplace-de Rham operator $\text{[math]}$ .

Another useful identity of Lagrange is

$\text{[math]}$

This is a special case of the multiplicativity $\text{[math]}$ of the norm in the quaternion algebra.

Ways to compute a cross product

Coordinate notation

The unit vectors i, j, and k from the given orthogonal coordinate system satisfy the following equalities:

i × j = k j × k = i k × i = j.

With these rules, the coordinates of the cross product of two vectors can be computed easily, without the need to determine any angles: Let

a = a₁i + a₂j + a₃k = (a₁, a₂, a₃)

and

b = b₁i + b₂j + b₃k = (b₁, b₂, b₃)

Then

a × b = (a₂b₃ − a₃b₂) i + (a₃b₁ − a₁b₃) j + (a₁b₂ − a₂b₁) k = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

Matrix notation

The coordinate notation can also be written formally as the determinant of a matrix:

$\text{[math]}$

The determinant of three vectors can be recovered as

det (a, b, c) = a · (b × c).

Intuitively, the cross product can be described by Sarrus's scheme. Consider the table

$\text{[math]}$

For the first three unit vectors, multiply the elements on the diagonal to the right (e.g. the first diagonal would contain i, a₂, and b₃). For the last three unit vectors, multiply the elements on the diagonal to the left and then negate the product (e.g. the last diagonal would contain k, a₂, and b₁). The cross product would be defined by the sum of these products:

$\text{[math]}$

Although written here in terms of coordinates, it follows from the geometrical definition above that the cross product is invariant under rotations about the axis defined by $\text{[math]}$ , and flips sign under swapping $\text{[math]}$ and $\text{[math]}$ .

Quaternions

Further information: quaternions and spatial rotation

The cross product can also be described in terms of quaternions, and this is why the letters i, j, k are a convention for the standard basis on $\text{[math]}$ : it is being thought of as the imaginary quaternions.

Notice for instance that the above given cross product relations among i, j, and k agree with the multiplicative relations among the quaternions i, j, and k. In general, if we represent a vector [a₁, a₂, a₃] as the quaternion a₁i + a₂j + a₃k, we obtain the cross product of two vectors 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.

Conversion to matrix multiplication

A cross product between two vectors (which can only be defined in three-dimensional space) can be rewritten in terms of pure matrix multiplication as the product of a skew-symmetric matrix and a vector, as follows:

$\text{[math]}$

where

$\text{[math]}$

also if $\text{[math]}$ is a result of a cross product:

$\text{[math]}$

then

$\text{[math]}$

This notation provides another way of generalizing cross product to the higher dimensions by substituting pseudovectors (such as angular velocity or magnetic field) with such skew-symmetric matrices. It is clear that such physical quantities will have n(n-1)/2 independent components in n dimensions, which coincides with number of dimensions for three-dimensional space, and this is why vectors can be used (and most often are used) to represent such quantities.

This notation is also often much easier to work with, for example, in epipolar geometry.

From the general properties of the cross product follows immediately that

$\text{[math]}$ and $\text{[math]}$

and from fact that $\text{[math]}$ is skew-symmetric it follows that

$\text{[math]}$

Lagrange's formula (bac-cab rule) can be easily proven using this notation.

The above definition of $\text{[math]}$ means that there is a one-to-one mapping between the set of $\text{[math]}$ skew-symmetric matrices, also denoted SO(3), and the operation of taking the cross product with some vector $\text{[math]}$ .

Index notation

The cross product can alternatively be defined in terms of the Levi-Civita tensor $\text{[math]}$

$\text{[math]}$

where the indices $\text{[math]}$ correspond, as in the previous section, to orthogonal vector components.

Mnemonic

The word xyzzy can be used to remember the definition of the cross product.

If

$\text{[math]}$

where:

$\text{[math]}$

then:

$\text{[math]}$

Notice that the second and third equations can be obtained from the first by simply vertically rotating the subscripts, x → y → z → x. The problem, of course, is how to remember the first equation, and two options are available for this purpose: either you remember the relevant two diagonals of Sarrus's scheme (those containing i), or you remember the xyzzy sequence.

Since the first diagonal in Sarrus's scheme is just the main diagonal of the above-mentioned $\text{[math]}$ matrix, the first three letters of the word xyzzy can be very easily remembered.

Applications

The cross product occurs in the formula for the vector operator curl. It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product.

The cross product can also be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics.

Given a point p and a line through a and b in a plane, all with z coordinate zero, then the z component of (p-a) × (b-a) will be positive or negative, depending on which side of the line p is.

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

Cross product as an exterior product

The cross product can be viewed in terms of the exterior product. This view allows for a natural geometric interpretation of the cross product. In exterior calculus the exterior product (or wedge product) of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors a and b, one can view the bivector a∧b as the oriented parallelogram spanned by a and b. We obtain the cross product by taking the Hodge dual of the bivector a∧b; this can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. In three dimensions only (because only in this case the dual of a vector is a bivector), the result is an oriented line element -- a vector (whereas, for example, in 4 dimensions the hodge dual of a bivector is two dimensional -- another oriented plane element). So, in three dimensions only, the cross product of a and b is the vector dual to the bivector a∧b: it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as a∧b has relative to the unit bivector; precisely the properties described above.

Cross product and handedness

When measurable quantities involve cross products, the handedness of the coordinate systems used cannot be arbitrary. However, when physics laws are written as equations, it should be possible to make an arbitrary choice of the coordinate system (including handedness). To avoid problems, one should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two vectors, one must take into account that when the handedness of the coordinate system is not fixed a priori, the result is not a (true) vector but a pseudovector. Therefore, for consistency, the other side must also be a pseudovector.

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

vector × vector = pseudovector

vector × pseudovector = vector

pseudovector × pseudovector = pseudovector

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

A handedness-free approach is possible using exterior algebra.

Higher dimensions

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. See seven dimensional cross product for the main article. The nonexistence of cross products of two vectors in other dimensions is related to the result that the only normed division algebras are the ones with dimension 1, 2, 4, and 8.

In general dimension, there is no direct analogue of the binary cross product. There is however the wedge product, which has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector. The cross product can be interpreted as the wedge product in three dimensions after using Hodge duality to identify 2-vectors with vectors.

One can also construct an n-ary analogue of the cross product in Rⁿ⁺¹ given by

$\text{[math]}$

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,Λ(v₁,...,v_n)) have a positive orientation with respect to (e₁,...,e_n+1). If n is even, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is odd, however, the distinction must be kept. This n-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.

The wedge product and dot product can be combined to form the Clifford product.

In the context of multilinear algebra, it is also possible to define a generalized cross product in terms of parity such that the generalized cross product between two vectors of dimension n is a tensor of rank n−2. This is a different concept than what is discussed above.

History

In 1843 the Irish mathematical physicist Sir William Rowan Hamilton introduced the quaternion product, and with it the terms "vector" and "scalar". Given two quaternions [0, u] and [0, v], where u and v are vectors in R³, their quaternion product can be summarized as [−u·v, u×v]. James Clerk Maxwell 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.

However, Oliver Heaviside in England and Josiah Willard Gibbs in Connecticut felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. Thus, about forty years after the quaternion product, the dot product and cross product were introduced — to heated opposition. Pivotal to (eventual) acceptance was the efficiency of the new approach, allowing Heaviside to reduce the equations of electromagnetism from Maxwell's original 20 to the four commonly seen today.

Largely independent of this development, and largely unappreciated at the time, Hermann Grassmann created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. William Kingdon Clifford combined the algebras of Hamilton and Grassmann to produce Clifford algebra, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product.

The cross notation, which began with Gibbs, inspired the name "cross product". Originally appearing in privately published notes for his students in 1881 as Elements of Vector Analysis, Gibbs’s notation — and the name — later reached a wider audience through Vector Analysis , a textbook by a former student. Wilson rearranged material from Gibbs's lectures, together with material from publications by Heaviside, Föpps, and Hamilton. He divided vector analysis into three parts:

"First, that which concerns addition and the scalar and vector products of vectors. Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function."

Two main kinds of vector multiplications were defined, and they were called as follows:

The direct, scalar, or dot product of two vectors
The skew, vector, or cross product of two vectors

Several kinds of triple products and products of more than three vectors were also examined. The above mentioned triple product expansion (Lagrange's formula) was also included.

References

id="CITEREFCajori1929">Cajori, Florian (1929), A History Of Mathematical Notations Volume II, Open Court Publishing, pp. p. 134, ISBN 978-0-486-67766-8, <[1]
- id="CITEREFWilson1901">Wilson, Edwin Bidwell (1901), Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs, Yale University Press, <[2]
  External links
  - Z.K. Silagadze (2002). Multi-dimensional vector product. Journal of Physics. A35, 4949 (it is only possible in 7-D space)
  - Real and Complex Products of Complex Numbers
  - Vector Product Calculator Online application to calculate the vector product of 3 element vectors
  - An interactive tutorial created at Syracuse University - (requires java)
  - W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).
  - Multiplication of Vectors
  Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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  In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator.
  ..... Click the link for more information.
  spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
  ..... Click the link for more information.
  Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.
  
  An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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  In mathematics, orthogonal, as a simple adjective, not part of a longer phrase, is a generalization of perpendicular. It means at right angles, from the Greek ὀρθός orthos
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  dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.
  ..... Click the link for more information.
  scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector.
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  J. Willard Gibbs
  
  (1839-1903)
  Born January 11 1839(1839--)
  New Haven, Connecticut, U.S.
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  algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring.
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  associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed.
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  In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.
  
  The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
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  dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.
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  See also orientation (rigid body).
  
  In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which are "negatively" oriented.
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  In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper
  ..... Click the link for more information.
  In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions.
  ..... Click the link for more information.
  two-form is another term for a bilinear form, typically used in informal discussions, or sometimes to indicate that the bilinear form is skew-symmetric.
  
  In differential geometry, a two-form refers to a differential form of degree two.
  ..... Click the link for more information.
  Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.
  
  An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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  perpendicular (or orthogonal) to each other if they form congruent adjacent angles. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.
  ..... Click the link for more information.
  
  right hand grip rule.
  In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3-D.
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  In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. The three-dimensional counterpart of a parallelogram is a parallelepiped.
  ..... Click the link for more information.
  angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
  ..... Click the link for more information.
  spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
  ..... Click the link for more information.
  In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1 (the unit length). A unit vector is often written with a superscribed caret or “hat”, like this (pronounced "i-hat").
  ..... Click the link for more information.
  perpendicular (or orthogonal) to each other if they form congruent adjacent angles. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.
  ..... Click the link for more information.
  In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper
  ..... Click the link for more information.
  Area is a physical quantity expressing the size of a part of a surface. The term Surface area is the summation of the areas of the exposed sides of an object.
  Units
  Units for measuring surface area include:
  square metre = SI derived unit
  
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  In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. The three-dimensional counterpart of a parallelogram is a parallelepiped.
  ..... Click the link for more information.
  In mathematics, anticommutativity refers to the property of a operation being anticommutative, i.e. being non commutative in a precise way. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence in physics: they
  ..... Click the link for more information.
  In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:
  4 • (2 + 3) = (4 • 2) + (4 • 3).
  
  ..... Click the link for more information.
  associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed.
  ..... Click the link for more information.

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Cross Product

Information about Cross Product

Definition

Examples

Example 1

Example 2

Properties

Geometric meaning

Algebraic properties

Lagrange's formula (triple product expansion)

Ways to compute a cross product

Coordinate notation

Matrix notation

Quaternions

Conversion to matrix multiplication

Index notation

Mnemonic

Applications

Cross product as an exterior product

Cross product and handedness

Higher dimensions

History

See also

References

External links

Units