YouTube Physics Explanations Shouldn’t Use the Right-Hand Rule

Popular explanations of physical phenomena like gyroscopes or magnetic fields often end up having to explain the right-hand rule to explain how rotational quantities add (say, by using the right-hand rule to convert angular momenta into vectors, then adding the vectors).

This is bad, not just because the right-hand rule is confusing, but because it leads people to wonder if the right-hand rule has some physical reality.  For example, see the comments on this youtube explanation of gyroscopes by the PhysicsGirl: there’s a lot of confusion over whether the right-hand rule is a fact of nature or a convention.

It turns out that it’s unnecessary to use the right-hand rule because it’s actually unnecessary to convert rotational quantities to vectors at all, and I think many people are unaware of this.

Let me explain by analogy with the vector case. Given a particle, how do you represent its linear momentum?  You take a vector whose direction is the same as the direction of the particle’s velocity, and whose magnitude is the particle’s mass times its speed.

A system of two particles has a linear momentum obtained by adding the linear momenta of the individual particles by putting the vectors head-to-tail.

Now, how do we represent the angular momentum of a particle x with respect to some base point O?  Usually, what you do is take the position vector \vec{r} of x with respect to O and the particle’s linear momentum vector \vec{p}, and define the linear momentum to be the cross product \vec{r}\times\vec{p}, which requires the right-hand rule.  Then, as before, the angular momentum of two particles is the sum of the angular momentum of the particles individually.

However, there is another thing you can do: instead of taking the cross product \vec{r}\times\vec{p}, which is a vector, represent it as a 2-dimensional object: an oriented parallelogram called \vec{r}\wedge\vec{p} which is in the same plane as \vec{r} and \vec{p} and whose area is |\vec{r}||\vec{p}|.

These add analogously to vectors: you add two oriented parallelograms by matching up edges of the same length and opposite orientation.  (Apologies for the poor handwriting: the squiggles inside the parallelograms are meant to be arrows indicating orientation.)



Two of these parallelograms are declared to be equal if they have the same (signed) area and live in the same plane: in that way, you can always add any two of them by reshaping one of them to have the right side length to add with the other.

And, this addition is still appropriate: under this definition, the angular momentum of the system of particles is the sum of the angular momenta of the individual particles, and it has all the properties you want: it’s unchanged unless the system experiences a net torque (also represented by a parallelogram), etc.

This gets rid of the arbitrariness and complexities of the right-hand rule, and also has the nice property that for a situation restricted to a plane, you don’t have to arbitrarily introduce a third dimension not part of the problem.

These parallelograms are actually called bivectors, and this is just a very small part of a much larger enterprise called geometric algebra that allows algebraic manipulation of higher-dimensional objects just as linear algebra allows the algebraic manipulation of vectors.  There’s a small cadre of physicists who think lots of physics should be redone with geometric algebra.

I’m not qualified to have an opinion on that, but I am pretty confident that using signed parallelogram addition in YouTube physics explanations would be clearer than using the right-hand rule.

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