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I've researched lots of places and still cannot wrap my head around the cross product.

The closest thing I have to an understanding of the cross product is that its a measure of how orthogonal two vectors are. But there are still lots of things about the cross product I dont understand.

Why do we make the product vector perpendicular to the plane formed by the multiplying vectors? How does that relate to the determinant of a 3x3 matrix?

I get that this is a very vague and general question, but If someone could help me understand why the cross product is defined the way it is and whats its meaning, or refer me to a good source, which again, I've tried reading several explanations, that would be highly appreciated. I'm happy to clear up anything that is unclear about my question

Arnold Doveman
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  • Cross product is used in various areas. It is not clear what you mean by intuitive but electromagnetic waves are a typical case where the direction of the energy propagation is determined by the cross product of the electronic vector with the magnetic vector. I am sure that you could find more applications of this type of usefulness of the cross product. – Moti Aug 19 '15 at 19:25
  • I assume you're not satisfied with the explanation that in physics the cross product is useful for computing the flux through a surface. Mathematically, the cross product is an alternating bilinear function of its inputs, which is such a simple thing that it's almost guaranteed to be relevant -- but hopefully other people will provide a more enlightening explanation. Note that $\text{det}(a,b,c) = a \cdot (b \times c)$. – littleO Aug 19 '15 at 19:26
  • Can I think about it in terms of spin. So if you have a x b, and the angle between them is very small that means the vector b is spinning slowly about the vector a as the axis, if the angle is close to 90 its spinning very fast.

    And the product vector shows the direction in which the vector b is moving at any point, as its spinning?

     – Arnold Doveman Aug 19 '15 at 19:30
  • @ArnoldDoveman I added an answer to the question that this was marked a duplicate of. See if it helps you understand the cross product at all. –  Oct 09 '15 at 00:05

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