I am sorry in advance for my English. I am studying math in a different language so I've tried to translate all terms as precisely as possible, but an edit to this question would be great. To the topic. Here's the task:
Using automorphisms method prove that "Dot product is positive" is not expressible in (R^2,E) where E stands for |x-y|==|x-z|
It's all about the 2 dimensional vectors. I have struggled on this task for almost three days for now, but no solution. I have tried reflecting the axis, doing so only with some vectors, taking normal vectors. Nothing worked for me
The predicate $E$ asserts that $y$ and $z$ are the same distance from $x$. So in particular, any bijection from $\mathbb{R}^2$ to $\mathbb{R}^2$ which preserves distance will be an automorphism of the structure $(\mathbb{R}^2; E)$.
So it's enough to find an isometry $f$ of $\mathbb{R}^2$ such that, for some $v, w\in\mathbb{R}^2$, we have $v\cdot w>0$ but $f(v)\cdot f(w)\le 0$.
HINT: Rather than looking at two vectors at once, think about a single vector all on its own. When is $v\cdot v$ not positive? Fix $v_0$ such that $v_0\cdot v_0$ is not positive, and some other $v_1$ such that $v_1\cdot v_1$ is positive. Can you think of an isometry of $\mathbb{R}^2$ which moves $v_1$ to $v_0$?