And the r hat you mentioned is essentially the same r that I use. So yes, for the L and r that I use, use unitary vectors, since you only want to figure out the direction. The magnitudes are down by actually doing the cross product. I hope this helps, although I know it's a bit late now, lol
The direction of the vector for a B magnetic field is figured out by knowing the direction of the flow of current and the direction of the vector that points towards the point analyzed (radius).
IE:
(direction of magnetic field ONLY, not for magnitude) B = L X r
Direction of magnetic...