In this problem there are three particles, with velocities $\vec{v_1}, \vec{v_2}$ and $\vec{u}$.
Relative to the particle moving at $u$, the velocities $v_1$ and $v_2$ are of equal magnitude and are perpendicular. Accordingly, show: $\left|\vec{u}-\frac{1}{2}(\vec{v_1}+\vec{v_2})\right|^2=\left|\frac{1}{2}(\vec{v_1}-\vec{v_2})\right|^2$
I couldn't work out whether to put this into components or not. I only got it to work for $v_1\not=v_2$
I have taken $\left|\vec{u}-\frac{1}{2}(\vec{v_1}+\vec{v_2})\right|$
To mean $\left(\vec{u}-\frac{1}{2}(\vec{v_1}+\vec{v_2})\right)$
Then I expand out the LHS that to get:
$|\vec{u}|^2-\vec{u}\cdot\vec{v_1}-\vec{u}\cdot\vec{v_2}+\frac{1}{4}(\vec{v_1}+\vec{v_2})^2$
Then I am left with something different to the RHS unless I make some certain assumptions.