Definition:Vector Cross Product

Definition
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space 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 $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of the vector space in question.

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 see

 * Equivalence of Definitions of Vector Cross Product


 * Lagrange's Formula
 * Cross Product is Anticommutative
 * Cross Product Not Associative


 * Definition:Dot Product