A mental lapse that sometimes occurs while trying to talk and write at the same time! Good catch - you are correct, it should be vdx as stated but not written. Luckily the subsequent line at 6:55 corrects the notation.
It is just a vector property that enables us to show that the streamfunction has a constant value along a streamline. We know that the velocity vector is aligned along the streamline, and the vector dr is also specified as being aligned along the streamline. The cross product of two parallel vectors (in this case V and dr) is also zero.
