Definition:Vector Cross Product/Definition 1

From ProofWiki
Jump to navigation Jump to search


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 $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ 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:

\(\displaystyle \begin{bmatrix} a_i \\ a_j \\ a_k \end{bmatrix} \times \begin{bmatrix} b_i \\ b_j \\ b_k \end{bmatrix}\) \(=\) \(\displaystyle \begin{bmatrix} a_j b_k - a_k b_j \\ a_k b_i - a_i b_k \\ a_i b_j - a_j b_i \end{bmatrix}\)

Also see