It seems reasonable that the dot product of two vectors is the same after they both have been rotated by the same amount.
The dot product of two orthogonal vectors is zero. The dot product of the two column matrices that represent them is zero.
Only the relative orientation matters. If the vectors are orthogonal, the dot product will be zero. Two vectors do not have to intersect to be orthogonal. (Since vectors have no location, it really makes little sense to talk about two vectors intersecting.)
Of course, this is the same result as we saw with geometrical vectors.