A vector of rotation can be defined as a direction along the axis of rotation with a magnitude proportional to the angle of rotation. I define two such vectors representing 360 degrees rotation along two axes that are at 90 degrees to each other. I then do a vector addition to obtain a vector at 45 degrees from the two axes and with a magnitude of 509 degrees of rotation.
But clearly if I rotate a thing once completely around one axis and then once completely around an orthogonal axis, the net result isn't a rotation of 509 degrees around an axis halfway between. What gives?
Correct. You cannot define a vector of rotation this way because such "vectors" do not commute under addition. Took me a while to realise what was wrong.
But it is. Just that people don't generally write it that way. If you mean angular position, that is. Probably because the resultant cannot be calculated as can linear velocity components. So it's not a conventional vector. Same goes for angular velocity - there's no resultant that makes physical sense, just 3 separate numbers.
Depends on your criterion for vector, I guess. If you can't do normal vector operations with it I'd call it something else. Anyway why do angular velocity vectors obey the rules of vector algebra whereas angle "vectors" don't?
I don't fundamentally know. Angular velocity vectors are in fact orthogonal by definition to the angle whose rate of change they represent. That would of course be a whacky relationship if applied somehow to linear position and velocity vectors.
They can be decomposed into components in whatever direction you like - it's how you explain Foucault's pendulum. They are ordinary vectors. It just struck me as strange that you can't define a similar vector for angles. After all, velocity and position are pretty closely related.
The answer seems to be that angle "vectors" that have only small differences between them do indeed follow the normal rules of vector algebra. And of course angular veloceties are made from angle vectors that differ infinitesimally.
Once again: For linear components, position and velocity vectors are collinear. For angular components, position and velocity vectors are orthogonal. I suspect this is the root of the issue.
I keep hearing about the Planck length, than which nothing is smaller, and the Planck time, than which nothing is quicker, but I rarely hear about the Planck mass (0.02 mg), which is derived in exactly the same way as the other two but is quite measurable and than which many things are lighter. It makes me wonder whether the great significance attached to the length and the time is justified.
When two black holes merge, the total area of the horizon(s) INCREASES because area and entropy are directly related and entropy always increases. Also, the total mass DECREASES because gravitational radiation removes energy from the system upon merger. Horizon radius is proportional to mass, and horizon area proportional to radius squared.
When one noodles all these facts together, one gets strict limits on the amount of areal increase and mass decrease. High school level maths and quite instructive!