Definition:Vector Cross Product

Definition
Let $\mathbf a$ and $\mathbf b$ be $3$-dimensional vectors, such that:


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

where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of the vector space in question.

Then the vector cross product, denoted $\mathbf a \times \mathbf b$, is defined as:
 * $\mathbf a \times \mathbf b = \begin{vmatrix}

\mathbf i & \mathbf j & \mathbf k\\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$

where $\begin{vmatrix} \ldots \end{vmatrix}$ can be understood as a determinant.

More directly:
 * $\mathbf a \times \mathbf b = (a_j b_k - a_k b_j)\mathbf i - (a_i b_k - a_k b_i)\mathbf j + (a_i b_j - a_j b_i)\mathbf k$

If the vectors are represented as column matrices:
 * $\mathbf a = \begin{bmatrix} a_i \\ a_j \\ a_k \end{bmatrix}, \mathbf b = \begin{bmatrix} b_i \\ b_j \\ b_k \end{bmatrix}$

we can express the dot product as:

Also see

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