Setting the radius to 1 doesn't really simplify a problem. Therefore, we can try to get the general solution for an arbitrary radius. Apparently the locus of this cylinder will be the set of points in the space the distance from which to the axis will be equal to the radius given. Considering the point (x,y,z) as the one that would satisfy the equation of this locus we would have:
If the given vector is the unit vector, it will obviously make the equation less bulky, by eliminating the denominator in the fraction.