I have a vector {f} and I want to define an {s} vector that is orthogonal to {f} vector in which the {f} vector lies within the circumferential plane and the {s} vector lies within the horizontal plane. I have attached an image of my calculation for defining the {s} vector, in this way the {s} vector lies within the horizontal plane and it is orthogonal to {f} vector, but the orientation of {s} is not what I want, I want the orientation of {s} to be exterior, for better illustration of the exterior orientation I have attached another image, in this image the blue vectors are {f} and the red vectors are {s}, as you can see in the image, most of the orientations of the {s} vector are not exterior (I have illustrated the exterior orientation in the image) and some of them are, I think the reason for that is the definition of the alpha variable for {s} vector, I have defined alpha some have that always have positive value and this is not correct since this variable can be negative. Can anyone please help me to define the exterior orientation for the {s} vector?
Any feedback would be highly appreciated.