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I have learned about the cross-product of vectors in both mathematical studies and physical ones but I am still curious as to how one can obtain an intuitive understanding of how two vectors create a third vector in the perpendicular direction to the plane of the constituents.

In physics, the cross-product has many useful applications such as in electro-magnetism with the "right-hand rule" or likewise with torque, but it still does not feel right as there is no component of any of the constituent vectors that could be in the direction of the cross-product.

It has been my understanding that in mathematics, common operations such as the cross-product are derived from intuition. Furthermore, the development of mathematics always appears to precede physics in that mathematical knowledge is applied to physics such as in the case of imaginary numbers. But due to my inability to find an intuitive understanding of how to derive the cross-product I am driven to wonder whether it was developed from observations of physical phenomena.

So, my question is; was there an intuitive approach to knowing that the cross-product is perpendicular to the constituent vectors, or was this developed out of necessity from physical observations?

Additionally, could it be the case that in non-Euclidean space or other systems, that the cross-product has a clearer line of derivation?

Note: I acknowledge that intuition is subjective so when I say, "I can not find an intuitive approach to understanding the cross-product" I mean that I can not find an approach relative to my own intuition and am hoping that someone who might have been in the same boat can help me learn.

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  • $\begingroup$ Physics preceed mathematics. Unless you are talking about pedagogy, in which of course you need you maths ready first. $\endgroup$
    – Trebor
    Jun 26, 2022 at 11:42
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    $\begingroup$ Does this answer your question? What is the logic/rationale behind the vector cross product? $\endgroup$
    – Trebor
    Jun 26, 2022 at 11:43
  • $\begingroup$ I did look at that link first but I was still grappling with why the cross-product is perpendicular? Since you mentioned the article though, it brings about another question of why the cross-product vector is a pseudovector despite it being tangible in the physical world? $\endgroup$
    – Johnson
    Jun 26, 2022 at 11:50
  • $\begingroup$ Pseudovector is not saying that the vector is false, imaginary or pseudo. It's just a name that says when you look at the mirror image, pseudovectors gain an extra minus sign. $\endgroup$
    – Trebor
    Jun 26, 2022 at 11:53
  • $\begingroup$ @Trebor, in response to your comment, when I say, 'mathematics preceded physics" I mean in the sense that mathematical tools and knowledge are often developed and then years later applied to physical situations. $\endgroup$
    – Johnson
    Jun 26, 2022 at 11:54

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In the end, a friend showed me a great YouTube video by 3B1B that gives one of the best explanations of the cross product. For those that may wish to see it; https://www.youtube.com/watch?v=BaM7OCEm3G0

Thank you to those that commented and helped me!

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