First, a "unit vector parallel to "canbe "perpendicular to " only if is perpendicular to itself. So first take the dot product of those two vectors and set it equal to 0 to find r.Thenfind the unit vector in that direction.

You could, of course, do it your way. Since you say you have "too many equations" and "too many unknowns", I suspect you are not using the fact that U= ai+ bj+ c, is parallel to i+ rj+ 2k and so must be of the form ci+ crj+ 2ck. You have only two unknowns and two equations: the fact that the vector is perpendicular to 2i+ 2j- k gives you 2c+ 2cr-c= 0 and that its length, . In the first equation "c" immediately cancels leaving a simple equation for r. So that boils down to what I said above.