I am wondering which research communities use the notation $\mathbf i$, $\mathbf j$, $\mathbf k$ for the three-dimensional unit vectors. The calculus textbook I have to use (Stewart) uses that notation. But I am a mathematician, so maybe there is a physics or engineering community out there where that notation is actually used.

Notations that I am well acquainted with include

  • $e_1$, $e_2$, $e_3$
  • $e_x$, $e_y$, $e_z$
  • $\mathbf e_1$, $\mathbf e_2$, $\mathbf e_3$
  • $\mathbf e_x$, $\mathbf e_y$, $\mathbf e_z$
  • ...

I have never seen $\mathbf i$, $\mathbf j$, $\mathbf k$ in any research paper ever. Can you point out a recent example?

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    $\begingroup$ This question doesn't seem to be about mathematics education since you're not interested in textbooks that use the notation. $\endgroup$ – Joel Reyes Noche Mar 4 '18 at 11:12
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    $\begingroup$ It is about mathematics education. I am interested in a textbook that uses that notation since I have to use it. I would like to understand what I am teaching to my students. $\endgroup$ – shuhalo Mar 4 '18 at 15:51
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    $\begingroup$ In my opinion the question asks for the use of these symbols in research papers, which takes this outside the context of mathematics education. If you want to know about the use of this notation in textbooks (other than Stewart) that would seem like an on-topic question. $\endgroup$ – mweiss Mar 4 '18 at 20:28
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    $\begingroup$ "I would like to understand" -- surely the problem isn't that you don't understand the notation. You just want to know whether that notation is common to particular communities. $\endgroup$ – mweiss Mar 4 '18 at 20:29
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    $\begingroup$ I'm voting to close this question as it asks about the use of a specific notation in research, which is (just slightly) off-topic for this site. It could easily be edited to bring it on-topic, for example by asking about textbooks rather than research papers. $\endgroup$ – mweiss Mar 5 '18 at 2:18

Search arxiv.org for "unit vectors i, j, k" and you will find examples of research papers where the notation is used. Many are from physics communities investigating phenomena in three-dimensional space. Clearly we should not expect to find this notation in research relating to general dimensionality.

In the image see "unit vectors i, j, k" at the bottom.

Search Field on arXiv

  • $\begingroup$ Can you please give me one such example? I really don't see it and would like to know. Thanks. $\endgroup$ – shuhalo Mar 4 '18 at 17:57
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    $\begingroup$ I don't think this is the place to include links to arXiv papers, but in my answer I placed a screen shot so you can see how to do the search. Pick some hits and search the pdf for "unit vector" and you will find examples, If you reorder by date, the most recent paper is titled "Lorentzian geometry of qubit entanglement." $\endgroup$ – user52817 Mar 4 '18 at 18:42
  • $\begingroup$ It might be of note that using the same procedure for "unit quaternions i j k" brings up papers in stable homotopy theory, differential geometry, and mathematical physics. $\endgroup$ – K B Dave Mar 4 '18 at 20:54
  • $\begingroup$ I only tried google scholar before but the arxiv searched indeed brought up a few examples. Good point. $\endgroup$ – shuhalo Mar 5 '18 at 2:06

I'm not sure this a research setting, but the cross product relations $i \times j = k$, $j \times k = i$, etc are nice for illustrating the connection between quaternions (where $ij=k, jk=i$, etc) and the cross product.

Specifically, the Lie group consisting of unit length quaternions has Lie algebra $\mathbb{R}^3$ with Lie bracket given by cross product.


This notation is completely standard in all Physics classes at the undergraduate level. I doubt you would find any Physicist anywhere who does not instantly recognize that notation -- but then, I would also have said the same thing about mathematicians, as the notation is completely standard in multivariable calculus at the undergraduate level in the United States.

  • $\begingroup$ It is completely unheard of in Europe, and I haven't seen it in mathematics from graduate level onwards. So it was natural to ask where the notation is used in practice. $\endgroup$ – shuhalo Mar 5 '18 at 4:42
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    $\begingroup$ It's certainly in widespread use in the UK. $\endgroup$ – mweiss Mar 5 '18 at 4:43
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    $\begingroup$ It's certainly quite common in Italy $\endgroup$ – Nicola Ciccoli Mar 6 '18 at 9:04
  • $\begingroup$ The question asks which "research communities" use this notation. While it is certainly true that this notation is standard in multivariable calculus classes in the US, it is equally true that not so many mathematicians use it in their research papers. $\endgroup$ – Dan Fox Mar 9 '18 at 6:32


Crystallographers sometimes in my experience. But, first page of a Google Scholar search gave 9/10 results from data science statistics, PCA and such. Some in an application of statistics like a specific medical paper, others just stats or computing in general). There was only one paper that was crystal-related (lattice vibrations, but in a physics context, not structure determination). So presumably space vectors or lattice vectors (perhaps not Cartesian).


(You may see slightly different specific results from your search, Google being Google, but no matter, still shows you some people using ijk)

  • $\begingroup$ Can you give me an example paper? The results that I see use ijk mostly as indices, or maybe I am just blind, or Google is deliberately fooling me. $\endgroup$ – shuhalo Mar 4 '18 at 15:49
  • $\begingroup$ I think you are right. My bad. $\endgroup$ – guest Mar 4 '18 at 16:19
  • $\begingroup$ P.s. I'm surprised you didn't complain about using 3 dimensions. See this link for amusement: youtube.com/watch?v=MZZPF9rXzes $\endgroup$ – guest Mar 4 '18 at 16:22

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