I have a vector $\mathbf{x} = (x_1,x_2)$, how do I represent the array of such vector?
Is it $\mathbf{x} \in \mathbf{X}$ where $\mathbf{X}$ = $\bigcup _{{1}}^{k}\mathbf{x}^s$ for $s \in (1,..k)$ length of $k$ vectors? Something seems off here!
Also I want to index a vector in array then is this correct way to represent the index ${\mathbf{x}^1}$? and the corresponding dimensional representation of this index shall be $(x_1^1,x_2^1)$?
Is the usage of "hat" appropriate here ? such as "$\hat{x}$"?
I'm not sure if by "array of vectors" you intend a single row with the vectors listed, or if you mean a rectangular array with the components as entries or what. But I think the most common thing one would do in a situation like this is to put them in a rectangular matrix and use double indices.
That is, you would have $m$ vectors with $n$ components that look like $(x_{i,1}, \ldots, x_{i, n})$ for $i\in \{1\ldots m\}$, so that the rows of your matrix are the original $n$-tuples.
I have no idea why one would use a hat decorator in this case.