TITLE: What is a parallel line?
QUESTION [9 upvotes]: We are learning vectors in class and I have a question about parallel lines and coincident lines. According to wikipedia a parallel line is: 

Two lines in a plane that do not intersect or touch at a point are called parallel lines.

But another reference says

Side by side and having the same distance continuously between them.

According to the above definition, you can have two parallel lines with a distance of zero but this contradicts the first definition. So which is it?
EDIT:
If it is the second definition then you can have two lines that are coincident and parallel right?

REPLY [6 votes]: It's always important, in situations like this, to recall the immortal words of Humpty Dumpty (via Lewis Carroll's Through the Looking Glass):

"When I use a word, [...] it means just what I choose it to mean --- neither more nor less."

Indeed, Wikipedia's discussion of "Equivalent Properties" to Euclid's Parallel Postulate notes four "common definitions" of the term parallel: (1) constant separation, (2) never meeting, (3) same angles where crossed by some third line, or (4) same angles where crossed by any third line. (I'd say that (3) and (4) are slightly-nuanced ways of conveying the common notion of parallel lines "pointing in the same direction".)
Observe how these four "common definitions" apply to your "distance of zero" (aka, "coincident") lines:

Constant separation holds ... provided you allow that "separation by distance zero" counts as "separation". (That, too, is a choice.)
Never meeting fails: coincident lines meet everywhere.
Same angles where crossed by some third line holds: coincident lines are going to make identical angles!
Same angles where crossed by any third line holds: as in 3.

That's either 3-to-1 in favor of "coincident lines are parallel", or a 2-2 split decision; so the answer to your question "So, which is it?" is: It depends.

Personally, I'm a big fan of the "never meeting" definition of parallelism; it just seems the cleanest formulation. Number of points in common? Zero! Neat and tidy, whereas the other definitions involve a lot of extra work, measuring infinitely-many distances or at least a couple of angles. But, doggonit, it's just so convenient in algebraic geometry to say "lines with the same slope are parallel", or in linear algebra to say "vectors with the same (or proportional) components are parallel". My favorite trigonometric diagram has a nifty mnemonic that "reciprocal functions appear as parallel segments": $\sin \parallel \csc$ and $\cos \parallel \sec$; the "same direction" interpretation allows me to include $\tan \parallel \cot$ (and also $1 \parallel 1$), and it keeps the mnemonic from breaking down on a technicality at $0^\circ$ and $90^\circ$ when pairs of these "parallel" segments coincide. In the latter cases, being a stickler about "zero points in common" can get in the way of what's important about the situation.
The point is: One can make the case that coincident lines are parallel, and one can make the case the they aren't. Every day, mathematicians use the word "parallel" in ways that favor one choice over the other, and they freely change their minds depending on context. What they don't do, however, is leave their audience guessing on the matter, the way Humpty does to Alice. If there's potential confusion over any definition, a good mathematician gives explicit clarification about "what I choose [the word] to mean".
A common way of barring coincident lines from the Parallel Club is to be very vigilant to use the word "distinct" to describe the lines in question; that's why you see so many answers here making a big deal about the word. Another strategy is perhaps to use phrasing such a "strictly parallel" ... the latter being akin to describing a company's profit as "strictly positive", because somewhere someone might be asking this version of your question:

Can "profit" be zero?