Cross product

The cross product (including vector product, vectorial product or outer product called ) is a link in the Euclidean vector space, which again assigns a vector in three-dimensional case, two vectors. To distinguish it from other products, in particular of the dot, it is written with a Malkreuz as a multiplication sign.

The cross product of the vectors and is a vector which is perpendicular to the plane spanned by the two vectors is plane and forms a right system with them. The length of this vector corresponds to the area of ​​the parallelogram, which is spanned by the vectors and.

In physics, the cross product occurs, for example in the calculation of the Lorentz force, as well as various parameters such as rotation torque, angular momentum, etc. on the Coriolis force.

Geometric definition

The cross product of two vectors and the three-dimensional space view is a vector which is orthogonal to and, and thus to and out of the plane spanned.

This vector is oriented so that form and in this order, a legal system, that is, and behave like the thumb, index finger and middle finger abducted the right hand ( right-hand rule).

The amount of indicates the area of ​​the parallelogram spanned by and. In terms of the area enclosed by angle and applies

In this case, respectively, and the lengths of the vectors and, and the sine of the angle enclosed by them.

In summary, therefore

Where the vector to the one and vertical unit vector, which adds them to a legal system.

Concept and notation

In different countries, different spellings are used for the vector product in part. In the English-and German-speaking countries, the notation is used for the vector product of two vectors, and usually, in France, however, the spelling is preferred. In Russia, the vector product is often or listed in the notation.

The spelling and the name of the outer product can be used not only for the vector product, but also for the link that associates two vectors called a bivector, see Grassmann algebra.

Component wise calculation

In right-handed Cartesian coordinates or in the real coordinate space with the standard scalar product and the standard orientation applies to the cross product:

A numerical example:

A rule of thumb for this formula is based on a symbolic representation of the determinant. Here one notes a matrix in which the first column contains the symbols and stand for the standard basis. The second column is formed by the components of the vector, and the third of those of the vector. This determinant is calculated according to the usual rules, for example by being developed in the first column

Or by using the rule of Sarrus:

With the Levi- Civita symbol is the cross product writes as

Properties

The cross product is bilinear, for all numbers, and and all the vectors and is

The cross product of a vector with itself or a collinear vector yields the zero vector:

The cross product is anticommutative. That is, with interchange of the vectors, it changes the sign:

For each vector, the following applies:

Here, the Malpunkt denotes the scalar product. With this condition, the cross product is uniquely determined.

Double cross product

Grassmann identity

The cross product is not associative. The Grassmann identity (after Hermann Grassmann ), also called Graßmannscher development kit, is for the repeated cross product of three vectors

The Malpunkte denote the scalar product. In physics, is often the spelling

Be used. According to this representation formula also BAC -CAB- formula is called.

Jacobi identity

In addition, the Jacobi identity is considered that the cyclic sum of repeated cross products vanishes:

Lagrange identity

For the scalar product of two cross products

For the square of the norm obtained from this

Thus, the magnitude of the cross product of

As the angle between and is always between 0 ° and 180 °, is

Triple product

The combination of cross and dot product in the form

Is called the scalar triple product. The result is a number, which corresponds to the defined volume of the plane defined by the three vectors spats ( parallelepiped ).

Connection with Lie algebra

The cross product can be defined for any body for the vector space. This then forms the cross product of a Lie algebra.

Polar and axial vectors

In the application of the cross product of vector physical quantities, the distinction plays in polar vectors ( that is those who, like differences between two position vectors behave as speed, acceleration, force, electric field strength) and axial vectors ( which behave like rotation axes, for for example angular velocity, torque, angular momentum, magnetic flux density) of a roll. Polar vectors assigned to the signature ( or parity) 1, axial vectors, the signature -1.

With vector multiplication with a polar vector vectors change their signature: Is a polar vector, it is an axial; is an axial vector, as is a polar. With vector multiplication with an axial vector, however, the signature remains.

Cross product and nabla operator

In the vector cross product of the analysis is used together with the nabla operator, to indicate the rotational differential operator. Is a vector field at so is

Again, a vector field, the rotation of.

Formally, this vector field is thus calculated as the cross product of the nabla operator and the vector field. However, the expressions occurring here are no products but applications of the differential operator on the function. Therefore, the above calculation rules such as Grassmann identity in this case are not valid. Instead, apply for double cross products with the nabla operator specific calculation rules.

Cross product in Rn

The cross product can be used for arbitrary dimension to generalize. In this case, the cross product is not a product of two factors, but of factors.

The cross product of the vectors is characterized in that the following applies for each vector

In coordinates can be the cross product of the calculated as follows. It is the associated -th canonical unit vector. For vectors

Applies

Analogous to the above-mentioned calculation using a determinant.

The vector must be orthogonal. The orientation is such that the vectors form a legal system in this order. The amount of is equal to the -dimensional volume spanned by the Parallelotops.

For frequently results not a product but only a linear map

The rotation by 90 ° clockwise.

This is followed can also be seen that the component vectors of the cross product, including the result vector in this order - unlike accustomed from the - generally do not form a legal system; they occur only in real vector spaces with odd, even at the result vector is the component vectors a left-handed system. This is again because the base in spaces of even dimension is not the same as the base, which is by definition (see above), a legal system. Although a small change in definition would mean that the vectors in the first-mentioned order in always form a legal system, namely when in the symbolic determinant of the column of unit vectors would be set to the far right. This definition has, however, not enforced.

An even further generalization leads to the Grassmann algebras.