Is it possible to transform a series of vectors from $\mathbb{Z} \to \mathbb{Z}^{+}$ while preserving the relative order of dot products?

Question

Given in input a series of vectors in the Integer coordinate space e.g.:

$ v1=\langle -1, -1\rangle, v2=\langle 2 -9 \rangle, v3=\langle -1 , 32 \rangle, v4=\langle 1, 5 \rangle$

is it possible to transform them in the domain of positive integers numbers, in such a way that the dot product relative order is preserved?

i.e.

v1 $\bullet$ v2 < v3 $\bullet$ v4

v1 $\bullet$ v2 > v1 $\bullet$ v4

etc...

Thanks!

Edited

• Removed the constraint on Real Number because it was irrelevant for the concrete problem.

• Modified the question asking for relative order rather than exact same dot product

Answer 1

Fix a vector <$-1$,$1$>

Say $f$ be a function that maps from integer coordinate 2D space to the naturals that is order-preserving with respect to the dot product in the sense defined

<$-1$,$1$>.<$n$,$0$>=$-n$ (where . represents the usual dot product)

So clearly $f(<n,0>)$ is strictly less than $f(<n+1,0>)$ as <$-1$,$1$>.<$n$,$0$> is less than <$-1$,$1$>.<$n+1$,$0$> and f is order preserving in the sense defined in the question, and f(<$1$,$1$>) is a fixed positive number.

Also the range of $f$ is the natural numbers but we get an infinite decreasing sequence of natural numbers which is not possible, hence such an order preserving-map in the sense defined is not possible.