Let $x \in \mathbb{Q}^+$. Then $x$ can be expressed uniquely as a product over primes:

$$x = \prod_{p \text{ prime}} p^{\nu_p(x)}$$

where $\nu_p(x)$ is the p-adic valuation of $x$. Let $\nu(x) = \langle \nu_2(x), \nu_3(x), \ldots \rangle \in \mathbb{Z}^{\oplus \mathbb{N}}$. Then $\nu(x) + \nu(y) = \nu(xy)$ and $a \nu(x) = \nu(x^a)$. Thus the positive rationals can be viewed as a module where addition and multiplication of two module elements correspond to multiplication and exponentiation of their corresponding rationals. Suppose we define the following "inner product":

$$\nu(x) \cdot \nu(y) = \sum_{p \text{ prime}} \nu_p(x) \nu_p(y)$$

This gives rise to a notion of orthogonality, angles, lengths, and volumes between positive rationals. My question is this: Do this "inner product" and the aforementioned notions have any significance, from the perspective of number theory? Have they been studied before?

For example: if $x,y \in \mathbb{Z}^+$ then $\nu(x) \cdot \nu(y) =0$ iff $x$ and $y$ are coprime. Furthermore, if $x \in \mathbb{Z}^+$ then $\lVert \nu(x) \rVert_1 = \Omega(x)$ and $\lVert \nu(x) \rVert_0 = \omega(x)$.