Define cross product of two vectors. Show that cross product of two vectors anti commute.

Cross product of two vectors is a vector whose magnitude is equal to product of the magnitude of each vector and sine of angle between them and is directed along the normal to the plane containing two vectors. 
               
The direction of the cross product of two vectors, that is  is determined by the right hand screw rule or right hand thumb rule. 
The right hand screw rule states that when you place a screw perpendicular to the plane containing vectors  Rotate the cap of a screw from  to find the direction of  and  to find the direction of .
The direction in which the screw will advance gives us the direction of .
Therefore, by Right Hand Screw Rule, the direction of 
Now, 
                      and
 
So, cross product of two vectors anti-commute.