Definition:Vector Cross Product

Definition
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions:


 * $\mathbf a = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k$
 * $\mathbf b = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k$

where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis of $\mathbf V$.

Complex Numbers
The definition is slightly different when the vector space under consideration is the complex plane, as the latter is of $2$ dimensions.

This is a modification of the vector cross product in which the resulting product is taken to be the length of the hypothetical vector which would be considered as being perpendicular to the complex plane.

Also known as
The vector cross product is often called just the cross product when there is no chance of confusion with other types of cross product.

The term vector product can also sometimes be seen, but again this can be ambiguous.

Also see

 * Equivalence of Definitions of Vector Cross Product


 * Lagrange's Formula
 * Vector Cross Product Distributes over Addition
 * Vector Cross Product is Anticommutative
 * Vector Cross Product is not Associative
 * Definition:Dot Product