Is this simply true by definition (that is, taken as axioms?)
How would one to prove that for $||\vec{x}||=1$ and $||\vec{y}||=1$, if $(\vec{x},\vec{y})=0$, then $\vec{x}\perp\vec{y}$?
In other words, which property of Euclidean space is the cause for this fact?
In other words$^{2}$, if my question can be actually proven, then what is the furthest we can go (in terms of proving) until the relationship between angle and length in Euclidean space is taken as definition?
$(x,y) = \|x\|\|y\| \cos \theta$ if $x$ and $y$ are vectors in $\mathbb{R}^n$. Where $\theta$ is the smaller angle between $x$ and $y$ in the plane/space generated by $x$ and $y$.
Now it is zero means : $\theta$ is $\frac{\pi}{2}$ when the angle is allowed from $(0,\pi)$.