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I am getting stuck in a really easy problem in Statistical Mechanics that involves elastic collisions, it is really very shameful that I am getting stuck at such a simple thing, but from:
$$
\left\|\overrightarrow{v_{1}}\right\|^{2}+\left\|\overrightarrow{v_{2}}\right\|^{2}=\left\|\overrightarrow{u_{1}}\right\|^{2}+\left\|\overrightarrow{u_{2}}\right\|^{2}
$$
and
$$
\overrightarrow{v_{1}}+\overrightarrow{v_{2}}=\overrightarrow{u_{1}}+\overrightarrow{u_{2}}
$$
How can I get
$$
\left\|\overrightarrow{v_{2}}-\overrightarrow{v_{1}}\right\|=\left\|\overrightarrow{u_{2}}-\overrightarrow{u_{1}}\right\|
$$
I tried completing the square in the first equation like:
$$
\overrightarrow{v_{1}} \cdot \overrightarrow{v_{1}}+\overrightarrow{v_{2}} \cdot \overrightarrow{v_{2}}-2 \overrightarrow{v_{1}} \cdot \overrightarrow{v_{2}}=\left(\overrightarrow{v_{2}}-\overrightarrow{v_{1}}\right) \cdot\left(\overrightarrow{v_{2}}-\overrightarrow{v_{1}}\right)=\left\|\overrightarrow{v_{2}}-\overrightarrow{v_{1}}\right\|^{2}=\overrightarrow{u_{1}} \cdot \overrightarrow{u_{1}}+\overrightarrow{u_{2}} \cdot \overrightarrow{u_{2}}-2 \overrightarrow{v_{1}}
$$
$\overrightarrow{v_{2}}$
and then using the second equation to get:
$$
=\overrightarrow{u_{1}} \cdot \overrightarrow{u_{1}}+\overrightarrow{u_{2}} \cdot \overrightarrow{u_{2}}-2 \overrightarrow{v_{1}} \cdot\left(\overrightarrow{u_{1}}+\overrightarrow{u_{2}}-\overrightarrow{v_{1}}\right)
$$
but I cannot seem to be able to simplify this to
$$
\overrightarrow{u_{1}} \cdot \overrightarrow{u_{1}}+\overrightarrow{u_{2}} \cdot \overrightarrow{u_{2}}-2 \overrightarrow{u_{1}} \cdot \overrightarrow{u_{2}}=\left\|\overrightarrow{u_{2}}-\overrightarrow{u_{1}}\right\|^{2}
$$
Can someone help me with this? I am sure it is quite simple, but since I am stuck I am losing way too much time on this.

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