Can a coordinate axis theoretically be a vector, since it meets the condition of having a magnitude and direction? i read on a NASA educational web page recently that the $x$ and $y$ axes in a Cartesian coordinate system are actually vectors. How true is that?
 A: The canonical answer is no because $\infty$ is not a member of $\mathbb{R}$. The points in a plane can be mapped to the vector space $\mathbb{R}^2$ where each point $(x,y)$ is identified as a vector, but infinite points are specifically excluded.
Now, you could artificially put infinite vectors into this vector space as long as you are being careful and define things properly, since I don't think you would violate any of the axioms of the vector space. But if you defined the vector $\omega_x = (\infty,0)$, then the distributivity of scalar sums means you would have define $2\omega_x, 3\omega_x, \dots$ and so to me this would end up looking like the space $V \times V$, whereby all finite points, $P(x,y)$, are elements $((x,y),0)$ and you could try to identify the x-axis with the vector $(0,(1,0))$ and the y-axis with $(0,(0,1))$, for example. You would then have an "extended" plane whereby all points in the plane, including such infinite points, can be mapped to $V \times V$, but the space of all points in this "extended plane" is not a vector space itself because this mapping is not bijective to $V \times V$. Note that the axes would still not formally be vectors in this space without some interpretation for it, but at this point you are so divorced from $\mathbb{R}^2$ that any physics in this space would not be recognizable: there's no benefit in trying to do it unless it is required.
A: There is nothing wrong with calling the axes as infinite vectors$^1$, however there is nothing useful about it as well. Why? Mostly because of the undefined (or infinite) magnitude of the vectors along the axes. Thus if we really need to show the directions of the axes using vectors then using vectors of finite magnitude would be far more helpful. This is where the concept of basis vectors comes from. Using $n$ basis vectors, we can define the directions of $n$ axes and at the same time, represent any vector in $\mathbb R ^n$ space with those $n$ basis vectors. The most common example of basis vectors is the standard basis, which in $\mathbb R^3$ (or 3D space) are $\mathbf{\hat i}$, $\mathbf{\hat j}$ and $\mathbf{\hat k}$.

$^1$ I know it is arguable that axes cannot be defined as vectors since the axes are infinite, however, here I am talkin about the limiting case of an infinite vector. An example of such a limiting vector would be:
$$\vec{v}= \lim _{a\to 1^+} \frac{\mathbf{\hat i}}{a-1}$$
Here, we can use $\vec{v}$ to denote the x-axis. This kind of limiting definition helps in sidestepping the explicit mention of infinite magnitude.
A: 
Vectors are usually denoted on figures by an arrow. The length of the arrow indicates the magnitude of the vector and the tip of the arrow indicates the direction. The vector is labeled with an alphabetical letter with a line over the top to distinguish it from a scalar. We will denote the magnitude of the vector by the symbol |a|. The direction will be measured by an angle phi relative to a coordinate axis x. The coordinate axis y is perpendicular to x. Note: The coordinate axes x and y are themselves vectors! They have a magnitude and a direction. You first encounter coordinates axes when you learn to graph. So, you have been using vectors for some time without even knowing it! 

Source
This is ... not accurate.  There is no sense in which coordinate axes can be considered vectors without mutilating any decent definition of a vector beyond recognition.  Perhaps this was typed up by a high school intern - in any case, it's baffling to see on the grc.nasa.gov domain.
A: You can think in this case that $x$ and $y$ mark to independent coordinates of a point. But one can always argue that there is a position vector between the origin of the $\mathbb R^2$ space associated with that point, so there is map between points of   $\mathbb R^2$  and position vectors in the same space and in that sense you may see $(x,y)$ as a couple of coordinates or as the components of vectors. just $x$ you could say is a vector in one dimension only. 
