$$\begin{align}

v_1 &\ne \, v_2 \\

\vec{v_1} &= (0,v_1) \\

\vec{v_2} &= (0,v_2) \\

\vec{v_1}' &= (-V,v_1/\gamma) \\

\vec{v_2}' &= (-V,v_2/\gamma)

\end{align}$$

Agreed?v_1 &\ne \, v_2 \\

\vec{v_1} &= (0,v_1) \\

\vec{v_2} &= (0,v_2) \\

\vec{v_1}' &= (-V,v_1/\gamma) \\

\vec{v_2}' &= (-V,v_2/\gamma)

\end{align}$$

The question is whether $$\vec{v_1}'$$ and $$\vec{v_2}'$$ are parallel.

In your document, you determine the angle of vectors from the ratio of their components (note that this is not length dependent):

$$\tan(\theta(\vec{v_1}')) = \frac{-v_1}{\gamma V} \\

\tan(\theta(\vec{v_2}')) = \frac{-v_2}{\gamma V}$$

If the angles are not the same, the vectors are not parallel. Obvious.\tan(\theta(\vec{v_2}')) = \frac{-v_2}{\gamma V}$$

Now you suddenly prefer a new definition of parallelism:

Let's apply that definition to $$\vec{v_1}'$$ and $$\vec{v_2}'$$:Tach said:two vectors are parallel if their unit vectors (versors) are identical.

$$\begin{align}

\vec{v_1}' &= (-V,v_1/\gamma) \\

\vec{v_2}' &= (-V,v_2/\gamma) \\

\hat{v_1}' &= \frac{\vec{v_1}'}{\|\vec{v_1}\|} \\

&= \left(\frac{-V}{\sqrt{V^2 +(v_1/\gamma)^2}}, \frac{v_1}{\sqrt{(\gamma V)^2 +v_1^2}}\right)

\hat{v_2}' &= \frac{\vec{v_2}'}{\|\vec{v_2}\|} \\

&= \left(\frac{-V}{\sqrt{V^2 +(v_2/\gamma)^2}}, \frac{v_2}{\sqrt{(\gamma V)^2 +v_2^2}}\right) \\

v_1 &\ne \, v_2 \\

\hat{v_1}' &\ne \, \hat{v_2}'

\end{align}$$

\vec{v_1}' &= (-V,v_1/\gamma) \\

\vec{v_2}' &= (-V,v_2/\gamma) \\

\hat{v_1}' &= \frac{\vec{v_1}'}{\|\vec{v_1}\|} \\

&= \left(\frac{-V}{\sqrt{V^2 +(v_1/\gamma)^2}}, \frac{v_1}{\sqrt{(\gamma V)^2 +v_1^2}}\right)

\hat{v_2}' &= \frac{\vec{v_2}'}{\|\vec{v_2}\|} \\

&= \left(\frac{-V}{\sqrt{V^2 +(v_2/\gamma)^2}}, \frac{v_2}{\sqrt{(\gamma V)^2 +v_2^2}}\right) \\

v_1 &\ne \, v_2 \\

\hat{v_1}' &\ne \, \hat{v_2}'

\end{align}$$

Therefore by your unnecessary definition, $$\vec{v_1'}$$ and $$\vec{v_2}'$$ are still not parallel.

Tach, do you really want to argue this? It seems like you're letting your desired result get in the way of clear reasoning.