K^3/Cross product/Direct/Definition

Let ${\displaystyle {}K}$ be a field. The operation on ${\displaystyle {}K^{3}}$, defined by

${\displaystyle {}x\times y={\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\times {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}:={\begin{pmatrix}x_{2}y_{3}-x_{3}y_{2}\\-x_{1}y_{3}+x_{3}y_{1}\\x_{1}y_{2}-x_{2}y_{1}\end{pmatrix}}\,,}$

is called the cross product.