The cross product of two vectors is not commutative because:

$\left|\overrightarrow{A}\times \overrightarrow{B}\right|=AB\sin \theta$ 

$\overrightarrow{A}\times \overrightarrow{B}=\hat{n}AB\sin \theta$

$\overrightarrow{B}\times \overrightarrow{A}=\hat{n}AB\sin \left(-\theta \right)$

$\overrightarrow{B}\times \overrightarrow{A}=-\hat{n}AB\sin \theta =-\overrightarrow{A}\times \overrightarrow{B}$

$\sin \theta =-\sin \left(-\theta \right)$

Direction of cross product is given by right hand thumb rule