Before revision: We can define the two unit vectors and the arbitrary vector in terms of their Cartesian coordinates in the old reference frame: , where and are the unit basis vectors of the old coordinate system and is the angle between the and unit vectors (i.e., the angle between the two reference frames).
After revision: We can define the two unit vectors and the random vector in terms of their Cartesian coordinates in the old reference frame: , where and are the unit basis vectors of the old coordinate system and is the angle between the and unit vectors (i.e., the angle between the two reference frames).