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I am honestly surprised I hadn't noticed this a while ago. It occurred to me while I working on some stuff involving finding the perpendicular vector to two others in 3 dimensions to form a coordinate frame on a parametric curve (apparently it is differential geometry) that the cross product had the same name as the set theory operator.

Why is this? A cross product to me seems like generating a set consisting of all ordered pairs of two sets whereas the other cross product takes two vectors and returns a third perpendicular one. Was there a reasoning behind this nomenclature? The fact that both deal with non-numerical objects (and a vector triple does form a basis for 3-dimensions) leads me to believe both are either the same thing (in some weird lets-define-vectors-with-set-theory way) or they both represent the same concept for both sets and vectors.

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    $\begingroup$ Entirely different meanings. Just reusing symbols in different contexts. $\endgroup$
    – copper.hat
    Oct 12, 2016 at 5:43
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    $\begingroup$ Can you give a source that calls the cartesian product of sets a "cross product"? Even if that's a thing, the symbol $\times$ is a cross, so... $\endgroup$
    – anon
    Oct 12, 2016 at 5:44
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    $\begingroup$ It's a Cartesian product of sets, not a cross product surely. $\endgroup$
    – SchrodingersCat
    Oct 12, 2016 at 5:46
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    $\begingroup$ The word "cross product" appears nowhere on the Wikipedia page for Cartesian product. The MathWorld entry parenthetically says "cross product" is another name, but I don't know what its source is. The google search results for "cross product sets" is not very suggestive. So your usage is niche. $\endgroup$
    – anon
    Oct 12, 2016 at 5:49
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    $\begingroup$ Oh...then perhaps I shouldn't mention that $ \times $ can symbolize the direct product of two groups! $\endgroup$
    – amWhy
    Dec 17, 2016 at 0:35

1 Answer 1

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Referring to the set theory cross product by the words "cross product" is technically wrong. It's officially called the "cartesian product." Some people just call it the cross product. In general, confusion should not arise. Ambiguities like this are often tolerated in mathematics, when technically, to be rigorous, such abuse of notation and terminology should be intolerable. In practice, however, it happens, and doesn't really cause problems because everyone knows what is meant. You'll see what I mean when you get into abstract algebra and all of a sudden, 8 different operations are denoted by the same symbol in the same problem just because it's too hard to write it the correct way by hand.

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    $\begingroup$ "Cartesian" should be capitalized because it comes from the name of a person. $\endgroup$
    – user856
    Oct 12, 2016 at 5:54
  • $\begingroup$ Whether it's regional or not; the important thing is seeing the ambiguity and knowing that the vector cross product and set cross product are unrelated. The term cross product is thrown around somewhat frequently in place of the cartesian product; and is used interchangeably often within math departments. However, a google search of the phrase "cross product" turns up no results referring to sets. They all refer to vectors. $\endgroup$
    – anonymous
    Oct 12, 2016 at 6:00
  • $\begingroup$ Also, there is another term for the cross product of vectors as well. It's often referred to as the "vector product." $\endgroup$
    – anonymous
    Oct 12, 2016 at 6:02
  • $\begingroup$ I had a book once that used the direct sum symbol from higher algebra as the symbol for the disjunctive union (aka symetric difference) between two sets. Like I said, it's a common practice; however, it's understood that to be logical and rigorous, everything within mathematics should have it's own name and symbol. In practice though, the ambiguity is tolerable and convenient. It's no big deal. $\endgroup$
    – anonymous
    Oct 12, 2016 at 6:06
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    $\begingroup$ @Rahul Occasionally a named thing will become decapitalized even when it's named after a person. For example, an "abelian group" is often not capitalized despite being named after Niels Henrik Abel. $\endgroup$
    – Alexis Olson
    Oct 12, 2016 at 7:33

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