Why we never use the product between vectors like between elements of direct groups product? We can say, that any field $\mathbb{K}$ -- $1$-dim vector space on itself: $\mathbb{K}_{\mathbb{K}}$. So any vector of one another finite-dimensional vector space $V_{\mathbb{K}}$, after choosing the some basis can be represented as the element of isomorphic space $\mathbb{K}_{\mathbb{K}}^{n} = \prod_{i=1}^n\mathbb{K}_i$, where $n$ -- dimension of $V_{\mathbb{K}}$. But we can determine operations on elements of $\mathbb{K}_{\mathbb{K}}^{n}$ like on the direct group product: $(x_1,\dots,x_n) + (x'_1, \dots, x'_n) = (x_1 + x'_1, \dots ,x_n + x'_n)$ and similar for second field operation: $(x_1,\dots,x_n) \times (x'_1, \dots, x'_n) = (x_1 \times x'_1, \dots ,x_n \times x'_n)$.
But usually we doing it only with one field operation $+$. Why? 

Because if $\mathbb K$ is a field, $\mathbb K^n$ remains a $\mathbb K$-vector space but may not be a field (with the same addition). For example, $\mathbb R$ is a field, but $\mathbb R^n$ can't have a field structure (with the classical addition) for $n=3$ and $n\geq 5$. Even when $n=4$ the field is not commutative. See for example the Wiki page on division $\mathbb R$-algebras
