Cross Product

In space $\mathbb{R}^3$, it is possible to transform the vector product operator into a linear application, that is, to give a matrix representation to the cross product, such that $[\mathbf{x}]_{\times}\mathbf{y} = \mathbf{x} \times \mathbf{y}$.

In the text, the matrix $3 \times 3$ associated with the cross product will be denoted by $[\mathbf{x}]_{\times}$. The form of this matrix, which is antisymmetric, is

$$[\mathbf{x}]_{\times} = \begin{bmatrix} 0 & -x_2 & x_1 \\ x_2 & 0 & - x_0 \\ - x_1 & x_0 & 0 \end{bmatrix}$$ (1.59)

where $\mathbf{x} = (x_0, x_1, x_2)^{\top}$. This matrix has zero determinant and maximum rank 2.