Angular displacement is not a vector

[spaghetti3451]

Homework Statement

The angular displacement θ, despite having a magnitude and a direction, cannot be treated as a vector. This is because θ does not follow the commutative law of vector addition.

Does the infinitesimal displacement dθ obey the commutative law of addition and hence qualify as a vector? If so, how is the direction of dθ related to the direction of ω?

Homework Equations

The Attempt at a Solution

That θ does not follow the commutative law of vector addition can be proved easily using an example of two rotations of some object.

What I cannot prove is whether the the infinitesimal displacement dθ obeys the commutative law of addition. Any help on that would be greatly appreciated.

 

[Opus_723]

You know that θ does not follow the commutative rule for vectors. This is easy to show, since you can rotate an object, say, 90° along one axis and then 90° around another axis, and see that the results are different depending on the order in which you perform the rotations.

So, knowing how to test for the commutative rule, try this test again with smaller and smaller θ. How different are the final positions for 45°? 15°? 1°? What would be the difference for an infinitesimally small value of θ?

It's easiest to do this with some object that's simple but has clearly different dimensions, like a book. That way you can compare orientations very easily.