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The vector notation in physics is a very compact and easy way to write things down, and according to Feynman it also saves print. When exactly did scientists realize that they were summarizing things with vector notation?
Recap: I know that Oliver Heaviside introduced vector calculus in the 20th century; I am asking about the vector notation in Newton's law, for example, or four-vectors, for that matter.
Vector notation was introduced to physics in the end of 19-th century. I am not 100% sure, but this is associated in my memory with J. Willard Gibbs (Wikipedia confirms this, and also mentions Heaviside). 3-dimensional vectors were preceded by quaternions, and there was even a discussion what is better: vectors or quaternions.
Maxwell in his Treatease on Electricity and Magnetism (1873) did not use vectors, and the standard mathematical textbook for physicists (Thomson and Tait, first edition 1867) does not mention them. Neither all multiple subsequent editions do.
As it is still not common knowledge, I will mention that vector notation is out of date. It was superseded by the formalism of differential forms of E. Cartan (invented in 1920-th). But differential forms still did not penetrate in undergraduate education, which seems very strange to me. And many (not all!) physicists continue to use vector notation. Only a few of US universities teach differential forms to undergraduates.
EDIT. To those who do not believe what I wrote in the last paragraph, I recommend to look in the undergraduate textbook which is used in Harvard: Bamberg and Sternberg, A course in mathematics for students in physics, vol. 2, chapter Maxwell equations. They explain by the way how notation evolved, from Maxwell's notation to "vector calculus" to "tensor calculus" to modern language of differential forms.
