Cross Product

In mathematics, the cross product, vector product, or Gibbs' vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering.

If either of the vectors being multiplied is zero or the vectors are parallel then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular for perpendicular vectors this is a rectangle and the magnitude of the product is the product of their lengths. The cross product is anticommutative, distributive over addition and satisfies the Jacobi identity. The space and product form an algebra over a field, 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 Euclidean space, but unlike the dot product, it also depends on the choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of n − 1 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.

Read more about Cross Product:  Definition, Names, Cross Product As An Exterior Product, Cross Product and Handedness, History

Other articles related to "cross product, cross products, product, products":

Cross Product - History
... Lagrange introduced the component form of both the dot and cross products in order to study the tetrahedron in three dimensions ... Sir William Rowan Hamilton introduced the quaternion product, and with it the terms "vector" and "scalar" ... Given two quaternions and, where u and v are vectors in R3, their quaternion product can be summarized as ...
Seven-dimensional Cross Product - Rotations
... In three dimensions the cross product is invariant under the group of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of x × y under the rotation ... true in seven dimensions that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7) ...
vectors" class="article_title_2">Levi-Civita Symbol - Applications and Examples - Vector Cross Product - Cross Product (two Vectors)
... coordinate system using an orthonormal basis), their cross product can be written as a determinant hence also using the Levi-Civita symbol, and more simply In Einstein notation ...
Cross Product (disambiguation)
... The cross product is a product in vector algebra ... Cross product may also refer to Seven-dimensional cross product, a related product in seven dimensions A product in a Künneth theorem A crossed product in von Neumann algebras A Cartesian product in ...
Seven-dimensional Cross Product - Coordinate Expressions
... To define a particular cross product, an orthonormal basis {ej} may be selected and a multiplication table provided that determines all the products {ei × ej} ... other unit vectors, allowing many choices for each cross product ... multiplication table from the one in the Introduction, leading to a different cross product, is given with anticommutativity by More compactly this rule ...

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