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we know that vector products are non-commutative. How can we find the right direction? e.g consider magnetic force direction i-e., F= q V × B. Why we can not write F= q B × V? How can we find the right order?

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  • $\begingroup$ Is experimentally a sufficient answer? $\endgroup$ Nov 18, 2019 at 3:11
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    $\begingroup$ You can derive this force (more generally Lorentz force) using Lagrangian mechanics, considering a potential of the form $V=q\phi - q \vec{A} \cdot \vec{v}$. $\endgroup$ Nov 18, 2019 at 3:19
  • $\begingroup$ @PabloNavarrete kindly what are you people saying? I didn't get it. $\endgroup$ Nov 18, 2019 at 3:26
  • $\begingroup$ @PabloNavarrete you're still left with having to choose an orientation for the cross product, which amounts to choosing a sign for $\epsilon_{0123}$ $\endgroup$ Nov 18, 2019 at 4:20
  • $\begingroup$ See "en.wikipedia.org/wiki/Right-hand_rule" - there's an image on right. $\endgroup$ Nov 18, 2019 at 6:20

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The “right-handed” direction of the cross product is just a convention, related to the biological accident that more people are right-handed than left-handed. You could equally well use a left-hand rule and the Lorentz force would be $\mathbf F=q\mathbf B\times\mathbf v$. There is no correct order, just a conventional handedness and a conventional order.

By the way, the direction of the magnetic field could also be taken to be the reverse of the conventional direction, since the Biot-Savart law expresses it as another vector product. But when computing the force between two currents, the arbitrariness in these two vector products “cancel” and give a non-arbitrary direction for the force: parallel currents attract and anti-parallel currents repel. Here is where the math has to match the physics, with no ambiguity in directions.

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  • $\begingroup$ I thought the cross product was a defined operation, and the right hand rule is just a nice way to determine the direction of that product. Or is it actually mathematically a "right-hand cross product"? $\endgroup$ Nov 18, 2019 at 12:48
  • $\begingroup$ @AaronStevens I’m not sure what you’re asking. A common definition, such as in Wikipedia, is “The cross product $\mathbf a \times \mathbf b$ is defined as a vector $\mathbf c$ that is perpendicular (orthogonal) to both $\mathbf a$ and $\mathbf b$, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.” So it is defined to use a right-hand rule, but it could have been defined to use a left-hand rule. $\endgroup$
    – G. Smith
    Nov 18, 2019 at 16:24
  • $\begingroup$ Right. I suppose it depends on where you start with the definition. $\endgroup$ Nov 18, 2019 at 16:32

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