Definition:Vector Cross Product/Definition 1

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 $\mathbf V$.

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 $\begin {vmatrix} \ldots \end {vmatrix}$ is interpreted as a determinant.

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

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 vector cross product as:

Also see

 * Equivalence of Definitions of Vector Cross Product