"u,v\in\mathbb{R}^n$ such that $u\cdot v=2$. For which $k$ does $\| u + Kv \| = \| Ku - v \|$ apply?

Question

The full question is : There are $u,v$ vectors from $\mathbb R^n$ and the scalar value of $u\cdot v = 2$.

What are the rational $K$ values so that the following equation applies: $\| u + Kv \| = \| Ku - v \|$.

I tried to raise the power of the equation by 2 and tried to solve it with roots equation. I'm really clueless about what to do next so any help would be great!

Sorry if I wrote something that is not understandable so if is there anything not clear about my question please ask:)

Edit - I didn't notice in the question that the vectors $U + V$ are unit vectors. That solved for me the question because i just putted insead of $||V||^2$ and for $||U||^2$ the value of 1 that helped me solve the equation and get the $k$ value of 0. The question can be closed.

BTW - I will check the link below for my further questions to be in the format needed. Thanks again!

Thanks

Answer 1

Hint: $\|x\|^2 = x\cdot x$. Use this to expand $\| u + Kv \|^2 = \| Ku - v \|^2$.