# Non-commutative property of vector cross product

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?

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

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.
• @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. Nov 18, 2019 at 16:24