I’m pretty sure that the difference is just that you get sqrt(2)/2 vs sqrt(2)/2 in the middle instead of .5 vs .5
The dot product of 2 unit vectors is equivalent to the cosine of the angle, the magnitude of the cross product is equivalent to the sine










I would guess that by “geometrically sum” they meant the magnitude (I don’t see what else they could have meant). I’m pretty sure I’ve heard it used in that way before although I don’t think it’s a standardized term