In the texbooks I have for linear algebra, the scalar products are only introduced for $\mathbb R$ and $\mathbb C$ vector spaces, that lead me to following question:
$W:=span(1,\sqrt{2}) \subset \mathbb C$ is a two dimensional $\mathbb Q$-vector space in $\mathbb R \subset \mathbb C$.
$V:=span(1,\sqrt{2},i) \subset \mathbb C$ is a three dimensional $\mathbb Q$-vector space in $\mathbb C$.
Is there a "useful" scalarproduct on $V$ or $W$ as $\mathbb Q$-vector spaces?
$\mathbb{R}$ and $\mathbb{C}$ are isomorphic vector spaces over $\mathbb{Q}$ of infinite uncountable dimension. For any subspace of finite dimension of these spaces we can define a scalar product such that a given basis of this subspace is orthogonal.
E.g., for your $V=span\{1,\sqrt{2},i\}$, any vector $v \in V$ can be expressed as a linear combination $ v=v_1+v_2\sqrt{2}+v_3i$, with $v_1,v_2,v_3 \in Q$ and we can define the scalar product of two vectors $v,u$ as: $$ \langle v,u\rangle=v_1u_1+v_2u_2+v_3u_3 $$
Clearly, for a subspace of infinite ( possibly uncountable) dimension, this definition is not useful.