What are the advantages, disadvantages and applications of writing vectors as row matrix, column matrix and sum? I am asking about three very common ways to write a vector $\mathbf{V}$ with $x$-component $A$, $y$-component $B$ and $z$-component $C$:
$$\begin{align}
\mathbf{V}&= A\mathbf{\hat i} + B\mathbf{\hat j} + C\mathbf{\hat k} \tag{1}  \\
&= \begin{bmatrix} A & B & C \end{bmatrix} \tag{2} \\
&= \begin{bmatrix} A \\ B \\ C \end{bmatrix} \tag{3} \\
\end{align}$$
What are the advantages, disadvantages and applications of notating a vector in manners $(1)$, $(2)$ and $(3)$?
My goal with this question is to have a reference for myself and others that explicitly answers these questions in a single, clear answer.
 A: First off, there's a big difference between row vectors and column vectors.  That's because formally, under the hood, row vectors aren't vectors at all.  They're covectors, also called linear forms or linear functionals, and what this means is that they actually represent a linear map $f:U\to \mathbb{R}$ in the same way an $n\times m$-dimensional matrix can be used to define a linear map $g: \mathbb{R}^m\to \mathbb{R}^n$ (where $U$ is the vector space containing $\bf{V}$).
This map is really easy to define, in normal Euclidean space it's just the dot product, or equivalently matrix multiplication:
\begin{equation*}
\begin{bmatrix}
   a & b & c
\end{bmatrix}
\begin{bmatrix}
    e \\
    f \\
    g
\end{bmatrix} = ae + bf + cg
\end{equation*}
Note that it doesn't make sense to multiply row vectors with row vectors, or column vectors with column vectors.  If you've seen physics, this is the motivation for the Bra-Ket notation: in the expression $\langle a | b \rangle$, $\langle a |$ is a row vector and $| b \rangle$ is a column vector.  Together, $\langle a | b \rangle = \langle a |\text{ } | b \rangle$.
This might seem a little silly right now, since covectors very clearly correspond to normal vectors, but that's not always the case (for example, in the infinite-dimensional spaces studied in functional analysis).
On the other hand, the form $$a{\bf\hat{i}} + b {\bf\hat{j}} + c {\bf\hat{k}}$$ is exactly the same as $$\begin{bmatrix}a\\ b\\ c\end{bmatrix}$$
I suppose the former notation is more succint in handwriting, but I've always found column vectors to be more illustrative -- you can easily tell the difference between vectors and scalars this way.  That said, the choice between the two is ultimately a matter of convenience.
