You reached that conclusion by assuming that:The angles made by the vectors $$\vec{v'_P(0)}$$,$$\vec{T'_1}$$ with the SAME direction (OS') are the SAME.
It's in the text.
I dispute that assumption, and ask you to demonstrate its validity.In frame S' that the ion velocities in S' are the prototypes for $$\vec{v'_P(0)}$$,$$\vec{T'_1}$$.