How to write the components of a vector from a set of vectors? Is any particular notation given below preferred in mathematical writing? Why?
Given a set of vectors $\vec{x}_1,\vec{x}_2,\dotsc,\vec{x}_N$, we can write an individual vector $\vec{x}_i$ as follows.
\begin{align*}
    \vec{x}_i & =\left(x_i^{(1)},x_i^{(2)}\ldots,x_i^{(n)}\right)^T \\
    \vec{x}_i & =\left(x_1^{(i)},x_2^{(i)}\ldots,x_3^{(i)}\right)^T \\
    \vec{x}_i & =\left(x_{i,1},x_{i,2}\ldots,x_{i,n}\right)^T \\
    \vec{x}_i & =\left(x_{1,i},x_{2,i}\ldots,x_{n,i}\right)^T
\end{align*}
Is any method considered better than the other or is it completely up to the author?
 A: In general, folks make a choice about whether they have "row vectors" or "column vectors". I think most math books favor column vectors, but this is just a vague impression from 50 years or so of idiosyncratic reading of math papers in certain areas. Generally, $a_{ij}$ denotes the entry in the $i$th row and $j$th column of a matrix. 
Sadly, for a vector $u$, $u_j$ denotes the $j$th entry, and if you're thinking of column vectors, then this is the $j$th row of an $n \times 1$ matrix. This is, perhaps, an argument in favor of row-vectors. 
$$
\newcommand{\bx}{{\mathbf x}}
$$
If you have a bunch of vectors like yours (I'm going to assume column vectors), it's pretty common to put them in a matrix, with $\bx_1$ as the first column, $\bx_2$ as the second, and so on. Why? Because, calling the matrix $X$, you have that 
$$
Xu
$$
for a column vector $u$, is a linear combination of the $\bx_i$s, namely it's
$$
u_1 \bx_i + u_2 \bx_2 + \ldots + u_n \bx_n.
$$
In that sense, it'd be nice if the $ij$ element of the matrix $X$ were the $i$th element of vector $\bx_j$. Sadly, as you see in the example for $u$, the way we select the $k$th element of a vector is to subscript it with a $k$. So in the matrix $X$, we have the $ij$ entry is $(\bx_j)_i$, which isn't very pretty. 
It makes things a little prettier if you list your original vectors with superscripts: $\bx^1, \ldots, \bx^n$, for then the $j$th entry of column $i$ is 
$$
\bx^i_j.$$
Sadly, this is entry $ji$ of the matrix, and if we follow the "rule" that for a matrix with a capital letter name like $B$, lowercase things like $b_{ij}$ denote it's entries, then this gives us
$$
x_{ji} = x^i_j
$$
which ...well, which is bound to lead to confusion until you get used to it. But it seems nicer, to me, than saying that 
$$
x_{ji} = (\bx_i)_j.
$$
One problem with superscripts is that they can be confused with exponents. In linear algebra, this is seldom a problem, but you should be at least aware of it as a possibility. 
A: I personally would prefer your first and third notation, because I don't like to change the position of an index. But the notation also depends of the context.

I also use $\vec{x}^1,\ldots,\vec{x}^n$ and then $\vec{x}^i=(x_1^i,\ldots,x_N^i)$. The most important is to stay consequent in your notation.
