Is there a symbol or notation for collinearity?

Mathematics Asked by Kantura on October 2, 2020

Let’s say points $a$, $b$ and $c$ are collinear.

Is there some way to express this using a symbol instead of writing out a sentence.

Note I’m not looking for a definition for collinearity.

I’m thinking something like: $vec{abc}$

For context I am doing a proof and I just want to say:

$therefore $ $a$, $b$ and $c$ are collinear” in concise mathematical language.

2 Answers

This something of a frame challenge. You ask if there is some notation to expression collinearity. My answer to that question is "Maybe, but it doesn't really matter."

A more pertinent question is "Should I use some notation to express collinearity?" This is a much more nuanced question and, while the answer is (again) "Mabye," there are some questions which you might ask yourself.

  1. Will notation make my thoughts more clear?

    The one and only job that notation is meant to perform is clear communication. If notation makes it easier to convey an idea, then use notation. If notation doesn't help to convey an idea, then don't use it. In many cases, it is faster and easier to simply use English to convey an idea—in these cases, stick with English.

  2. Is there standard notation in my field?

    I work in fractal geometry. In this field, many basic examples arise by composing functions over and over again. Given a collection of maps ${f_{j}}$ and a word $alpha = (alpha_1, dotsc, alpha_n)$, the notation $f^{alpha}$ denotes the composition $$ f^{alpha} = f_{alpha_n} circ dotsb circ f_{alpha_1}, $$ and $f_j^n$ is the $n$-fold composition of $f_j$ with itself. These notations are common in fractal geometry, but are likely to be misunderstood in broader contexts. Thus if I am writing for other fractals people, I'll us the superscript notation without much discussion; but if I am writing for a broader audience, I will either forgo the notation, or be very careful in how I use it.

    Similarly, if you know of a notation for collinearity which exists and is widely understood by your intended audience, use that notation. Otherwise, don't.

  3. How much space am I actually saving by introducing notation?

    Introducing notation (particularly nonstandard notation) introduces a certain amount of cognitive overhead—the reader must invest a certain amount of effort into understanding what you have written. Thus one should generally avoid notation, unless the idea being notated is (a) referenced over and over again and (2) can be expressed more succinctly in notation.

    By way of example, I recently edited a paper for a friend. On page 3, they introduced a sequence of values $a_n(k)$, and set $a = lim_{ntoinfty} a_n(3)$. The first time that $a$ was used in this manner was on page 15 of a 20 page paper. In this case, the notation didn't really help all that much, so I suggested that it be removed. A few more words needed to be added near the end, but we got rid of a paragraph of notation on the front-end, so, overall, things ended up better without notation.

In the case of collinearity, my feeling is that the answers to these questions are

  1. Probably not.
  2. I can't answer that, as I don't know what your field is. However, I suspect that the answer is "no".
  3. Very little. The phrase "$a$, $b$, and $c$ are collinear" does not take up much space.

In short, I suspect that you are better off simply using plain English here, and not introducing any notation. Unless you are seeing notation for this idea in papers which you are reading, just say it in words.

Answered by Xander Henderson on October 2, 2020

I have seen guys in aops (and the expert guys) use the notation for $A,B,C$ collinear as "$overline{A-B-C}$". I don't exactly know why, but this may serve an answer (plus point : more than three points on a line, just add that point)

Answered by user732848 on October 2, 2020

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