Cross product

In mathematics, the cross product or vector product is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space, and is denoted by the symbol. Given two linearly independent vectors and, the cross product, , is a vector that is perpendicular to both and, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.
The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel, or if either one has zero length, then their cross product is zero.
The cross product is anticommutative and is distributive over addition, that is,. The space together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of the space under consideration, but unlike the dot product, it also depends on a choice of an orientation of the space. The cross product is invariant under a rotation of the basis but is changed into its opposite by an odd permutation of the basis vectors. Therefore, the cross product is a pseudovector.
In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions 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, so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time.

Definition

The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by. In physics and applied mathematics, the wedge notation is often used, 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 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 defined by the formula
where
If the vectors a and b are parallel, by the above formula, the cross product of a and b is the zero vector 0.

Direction

The direction of the vector n depends on the chosen orientation of the space. Conventionally, it 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. 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, in general the cross product of two vectors is not a "true" vector, but a pseudovector.

Names and origin

In 1842, William Rowan Hamilton first described the algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors 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, and independently Oliver Heaviside, introduced the notation for both the dot product and the cross product using a period and an "×", respectively, to denote them.
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 is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities
A mnemonic for these formulas is that they can be deduced from any other of them by a cyclic permutation of the basis vectors. This mnemonic applies also to many formulas given in this article.
The anticommutativity of the cross product, implies that
The anticommutativity of the cross product also implies that
.
These equalities, together with the distributivity and linearity of the cross product, 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:
Their cross product can be expanded using distributivity:
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:
meaning that the three scalar components of the resulting vector s = s₁i + s₂j + s₃k = are
Using column vectors, we can represent the same result as follows:

Matrix notation

The cross product can also be expressed as the formal determinant:
This determinant can be computed using Sarrus's rule or cofactor expansion. Using Sarrus's rule, it expands to
which gives the components of the resulting vector directly.

Using Levi-Civita tensors

In any basis, the cross-product is given by the tensorial formula where is the covariant Levi-Civita tensor. That corresponds to the intrinsic formula given [|here].
In an orthonormal basis having the same orientation as the space, is given by the pseudo-tensorial formula where is the Levi-Civita symbol. That is the formula used for everyday physics but it works only for this special choice of basis.
In any orthonormal basis, is given by the pseudo-tensorial formula where indicates whether the basis has the same orientation as the space or not.

The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis.

Properties

Geometric meaning

The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides :
Indeed, one can also compute the volume V of a parallelepiped having a, b and c as edges by using a combination of a cross product and a dot product, called scalar triple product :
Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value:
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, 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 of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine.

Algebraic properties

If the cross product of two vectors is the zero vector, then either one or both of the inputs is the zero vector, or else they are parallel or antiparallel so that the sine of the angle between them is zero.
The self cross product of a vector is the zero vector:
The cross product is anticommutative,
distributive over addition,
and compatible with scalar multiplication so that
It is not associative, but satisfies the Jacobi identity:
Distributivity, linearity and Jacobi identity show that the R³ vector space together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO.
The cross product does not obey the cancellation law; that is, with does not imply, but only that:
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:
for some scalar t.
If, in addition to and as above, it is the case that then
As cannot be simultaneously parallel and perpendicular 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:
where is a rotation matrix with.
More generally, the cross product obeys the following identity under matrix transformations:
where is a 3-by-3 matrix and is the transpose of the inverse and is the cofactor matrix. It can be readily seen how this formula reduces to the former one if is a rotation matrix. If is a 3-by-3 symmetric matrix applied to a generic cross product, the following relation holds true:
The cross product of two vectors lies in the null space of the matrix with the vectors as rows:
For the sum of two cross products, the following identity holds: