How can I rotate a vector 90 degrees into a perpendicular plane, and then 15 degrees free from that plane?

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What I ultimately want is a vector, giving the direction of the green line in the image below, knowing only the position of the yellow and green dots.
To be more specific, it's angle can be random as long as it's endpoint ends up somewhere on the green-blue surface of the cylinder. So, 360° free around cylinder, and about 15° limited to the edges of the cylinder.

The cylinder is perpendicular to the line from the yellow and green dot.

Length is not important, only direction.

My main problem is I don't know how to go from vector Yellow to green dot, to any vector perpendicular to it.

PS None of these things are aligned on a x y z axis. That grid is not xyz, just to help visualize. enter image description here



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