TITLE: What does this vector notation really mean?
QUESTION [6 upvotes]: With regard to vectors, how is this (form 1):
$$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$
Different to this (form 2): 
$$\begin{bmatrix}1\ 2\ -1 \end{bmatrix}$$
I would think that the first set consists of magnitudes of the same variable (where simply the sum gives the magnitude), while the second refers to the coefficients of three different variables.  
In my book they take the magnitude of form 1 by $$\sqrt{1^2+2^2+(-1)^2}$$
That would indicate a set of vectors in different directions. So my question is; which of the two (rows or columns) give an indication of the variable or the vector direction / dimension?

REPLY [5 votes]: What is a vector? Well, it is not "something with direction and magintude". Instead a vector is an arbitrary element of a vector space. Of course this merely shifts the question to: What is a vector space? But that can be defined to be a structure with quite simple axioms.
That being said, triples of real numbers from a vector space (with suitable "obvious" operations), no matter which notation we use to write down sucxh a triple.
However, while we'd usually consider $\mathbb R^3$ as the set of triples of real numbers and write such triples in the form $(x,y,z)$, for the vector space $\mathbb R^3$, it is customary to use the column form, i.e. write vectors as in your form 1. The reason is that this makes most sense when introducing matrix multiplication (i.e., $Av$ where $A$ is a matrix and $v$ a vector). That would make a row vector a one-by-three matrix, an element of $\mathbb R^{1\times 3}$. But these two structures are isomorphic (i.e., essentiually indistinguishable).