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Technically, "relative velocity" is analogous to "relative slope".Freixas said:when you guys said "angle", you were talking about relative velocity.
Since in Euclidean trigonometry we have ##m=\tan\phi##, in special relativity we have ##(v/c)=\tanh\theta##.
- Relative slope would be like ##m_{rel}=\tan(\phi_B-\phi_A) =\frac{m_B-m_A}{1+m_B m_A}## (which is invariant of choice of grid orientation)... note that slopes don't "add" or "subtract".
- Relative velocity would be like ##v_{rel}/c=\tanh(\theta_B-\theta_A)=\frac{1}{c}\frac{v_B-v_A}{1-v_Bv_A}## (which is invariant of choice of inertial frame).
- In Galilean spacetime trigonometry, using Yaglom's Galilean trig functions (##\mbox{cosg}\eta=1##; ##\mbox{sing}\eta=\eta##, and ##\mbox{tang}\eta=\frac{\mbox{sing}\eta}{\mbox{cosg}\eta}=\eta##, relative velocity is ##v_{rel}/c=\mbox{tang}(\theta_B-\theta_A)=\frac{1}{c}\frac{v_B-v_A}{1-0v_Bv_A}=\frac{1}{c}(v_B-v_A)## (which is invariant of choice of inertial frame), where ##c## is any convenient velocity scale used to make the left-hand side dimensionless [it could be ##c=1 \rm m/s##].
Note that additivity [or subtractivity] of velocities in Galilean physics is an exceptional case... not the norm, when viewed in this generalized viewpoint (in comparison to Euclidean geometry and Special Relativity).
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