The line would not change its position by rotating the frame of reference.
The motion is the same irrespective of the reference point one takes.
For example the moon travels in the same orbit irrespective of where you see it from your location or me from another. (Considering both our frames are inertial or of the same state)
Mathematics offers us the freedom to choose how to describe the motion but the physics remains same.
Another example that would help in this case is a magnetic compass.....The magnetic needle would always lie in magnetic north south direction no matter how you turn, rotate , move etc.
Hope this helps.