Given two vectors $$ \vec{u} = \begin{pmatrix} x\\ y\\ z \end{pmatrix} ,\ \vec{v} = \begin{pmatrix} a\\ b\\ c \end{pmatrix} $$ We want to calculate the cross product, that is written $$ \vec{w} = \vec{u}\times\vec{v} $$ $$ \vec{w} = \begin{pmatrix} yc-zb\\ za-xc\\ xb-ya \end{pmatrix} $$
$$ 1)\ \vec{w} \cdot \vec{u} = \vec{w} \cdot \vec{v} = 0 $$ $$ 2)\ \vec{v}\times \vec{u} = -\vec{u} \times{v} $$