(I have only briefly skimmed the article, but ...)

Material such as this serves to remind the reader of what they already know, and contextualize it in a way that is relevant to this article. The article begins by telling the reader, "We all know what scalars and vectors are---here they are---but have you wondered what if ...", and taking the reader beyond. The introduction which you seem to find objectionable is only a small part of a much longer article.

In addition, different readers have different, mildly different notation styles. These introductory blobs inform the reader of the language in the article, and are essentially a friendly statement of definitions.

A third purpose is rhetorical: readers sometimes get stuck while reading text, and these parts of the article work as anchor points where they can loop back and "synchronize" with the writer.

I definitely agree but there's no way this is "English/American style". It's because people have grand plans of making their article/book accessible to everyone, and they start off explaining e.g. what a vector is, but pretty soon realise the don't want to write an entire vector algebra textbook so they seamlessly give up and jump straight into Stoke's theorem or whatever.

I read a Synopsys simulator manual that explained what double clicking was.

Probably. But why would so many people want to make their article/book accessible to everyone. Let's accept the fact that some topics, like vector algebra, are just not that interesting to everyone.

Vector algebra is at the heart of a fairly large industry - games. I don't think there can be enough of accessible and understandable content from that point of view.

Did this article at some point give up and jump straight into something too difficult?

There's an urban legend at Swedish universities that American text book authors get paid by the word, and that's why their books are so incredibly verbose.

When I studied physics at the university, our undergraduate textbooks where relatively thin volumes (e.g. Alonso&Finn I-III), whereas the engineering students had these massive textbooks (Young&Freedman etc.). When looking into these massive tomes, yes, they spend a lot of words, but also they apparently don't expect the reader to be able to apply calculus. So instead of showing, say, Coulomb's law, and assuming the reader is capable of integrating to calculate the interaction between a point charge and a line, they have a section describing the interaction between two point charges. Then an entirely separate section describing the interaction between a point charge and a line, with the formula as given without actually explaining that, hey, this formula, you know, results if we take the fundamental law and do this and that. Incredibly infuriating.

ha, thank you for your honest feedback. My intended audience is not PhDs or math majors, it is high school physics teachers, practicing engineers, programmers, precocious high school students, etc. Many of these people benefit from some definitions.

I include 3 sentences defining a scalar so that I could introduce the concept of grade.

I include a few sentences defining a vector because just read the comments here and you'll see there are many definitions of vector and I want to specifically call out the one I care about in this post. I am also using a nonstandard, color-based notation throughout the article so it is helpful to take a concept that people already know just to demonstrate my notation. This also lets me introduce the 3D interactive illustrations.

Did you read the rest of the article or were these two definitions so objectionable that you quit?

When you are writing to a general audience you can't assume what the reader knows or does not know.

I disagree -- whenever you write anything, you always assume something about your audience. In other words, there's no such thing as "general audience". E.g. in OP's article, people are already interested in Math, otherwise they wouldn't click to the article, yet alone got through the first paragraph.

Do you have an example of "Russian style explanation"?

Consider Kolmogorov’s entire Foundations of the Theory of Probability is 71 pages long [1].

[1] https://archive.org/details/foundationsofthe00kolm

... and that he revolutionised the understanding of turbulence in [1] which is four pages long.

[1] The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers, Dokl. Akad. Nauk. SSSR 30, 299-303. Reprinted in Proc. R. Soc. London A434, 9-13 (1991).