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I hope I'm right in saying that the cross-product, $\vec{A}\times \vec{B}$ of two vectors is defined by a right hand rule (e.g. if $\vec{A}$ points along the forefinger and $\vec{B}$ along the second finger, then $\vec{A} \times\vec{B}$ points along the thumb) if you're using a right handed co-ordinate system, but by a left hand rule if you're using a left handed coordinate system.

What motivates this difference in rules? I understand the concept of right and left-handed co-ordinate systems, but I don't understand why our definition of the cross-product itself should depend on the co-ordinate system. What would be wrong with continuing to define the cross product using a right hand rule, while using a left-handed co-ordinate system, and accepting the that its components will have different signs in a different co-ordinate system? [I'm using the cross product as an example of an axial vector.]

One more go at explaining my difficulty (mental block?) Take the magnetic Lorentz force, $\vec{F}=q \vec{v} \times \vec{B}$. What has the direction of $\vec{F}$ to do with co-ordinate systems? Isn't it fixed relative to $\vec{v}$ and $\vec{B}$ by a right hand rule (and various other conventions)?

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    $\begingroup$ In a left hand coordinate we have $\hat i\times\hat j=-\hat k$. You won't obtain that by using right hand rule. You'll need the left hand rule. $\endgroup$ – Diracology Jul 5 '17 at 13:18
  • $\begingroup$ Why complicate matters with conflicting hand-rules and coordinate systems, where you have to bother about the signs? It's much easier to keep things simple. Not sure if there's a deeper explanation for this. $\endgroup$ – Koo Zhengqun Jul 5 '17 at 13:25
  • $\begingroup$ @Koo Zhengqun It seems to me to complicate things to define a cross product differently according to co-ordinate systems. I've always thought of a vector as independent of co-ordinate system. I expect I'm missing something obvious. $\endgroup$ – Philip Wood Jul 5 '17 at 13:45
  • $\begingroup$ The cross-product is dependent upon the vector field/space in which it is defined. You cannot apply different rules to one and not the other. $\endgroup$ – honeste_vivere Jul 5 '17 at 13:56
  • $\begingroup$ There is a different relationship between the concepts. The coordinate system and cross product are cloaely related, in that you can describe the cross product in terms of coordinate system, or conversely, coordinate system in terms of cross product. You can use vectors to describe cross products as well, but you can also use vectors to describe lots of other things. On the other hand, you can't really use cross products to describe vectors. $\endgroup$ – Koo Zhengqun Jul 5 '17 at 14:01
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At some level your question just boils down to conventions: there are several sets of sign conventions that all give the right answers, so you just need to find one that you find conceptually satisfying and stick with it.

It might be helpful to point out that you can only ever directly measure polar vectors. Axial vectors can be thought of as just mathematical abstractions that only appear as intermediate steps in a physical process. For example, you are correct that in the Lorentz force law ${\bf F} = q{\bf v} \times {\bf B}$, the force is a physical quantity whose direction should not change sign under coordinate transformations. But remember that the magnetic field itself is physically determined by the Biot-Savart law (in the magnetostatic approximation; things get more complicated in the dynamic case but the parity transformation properties don't change): $${\bf B}({\bf x}) = \int d^3x'\ \frac{{\bf J({\bf x}')} \times \hat{\bf r}}{r^2},$$ where ${\bf r} := {\bf x} - {\bf x}'$. Therefore, the magnetic field is an axial vector and can't be directly measured; you can only ever observe its effect by taking another cross product to get a force via the Lorentz force law.

The punch line is that you can always expand out any physically measurable vector as being determined by an even number of cross products. For example, you can write the magnetic force on a particle as $${\bf F}({\bf x}) = q{\bf v} \times \int d^3x'\ \frac{{\bf J({\bf x}')} \times \hat{\bf r}}{r^2}.$$

So there are two different but physically equivalent ways to conceptualize a change in the handedness of your coordinate system. You can think of all "right-handed" cross products becoming "left-handed" cross products, in which case all axial vectors (like the magnetic field) will physically switch direction - but since they can't be physically measured, this has no consequences on observable physics. Or you can, as you prefer, think of cross products as being "physically" determined by the right-hand rule, in which case they don't switch direction because they don't care about your choice of coordinates. In this framework, the magnetic field does not switch direction under a coordinate inversion. Both conceptualizations lead to identical observable physics: since any directly observable vector is made up of an even number of cross products, it either picks up no minus signs or an even number of minus signs under a coordinate inversion. In either case, its direction is unchanged.

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    $\begingroup$ Thank you for this answer. Allowing a choice of viewpoints seems the way forward for me. I knew that my difficulties were more about language and conventions than about the Physics or even the maths. But many elementary presentations of this issue are dogmatic and, for me, confusing. $\endgroup$ – Philip Wood Jun 19 '18 at 18:00

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