The velocity  can be expressed in terms of the stream function  as
where  is the  rotation matrix corresponding to a  anticlockwise rotation about the positive  axis. Solving the above equation for  produces the equivalent form
From these forms it is immediately evident that the vectors  and  are
* perpendicular:  
* of the same length:  .
Additionally, the compactness of the rotation form facilitates manipulations (e.g., see Condition of existence).