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In Classical Mechanics angular velocity, angular acceleration, torque and angular momentum can be defined as vectors with clear advantages such as the possibility to use vector product to simplify expressions.

As someone who appreciates the symmetry between translational and rotational dynamics writing angular velocity as the derivative of the angle seems somewhat elegant to me, however this is not accurate when using vectors. This could be solved by defining an "angle vector". Why has this not common? Wouldn't it work?

I can imagine $\vec{\theta}$ perpendicular to the plane the angle lies on and with magnitude equal to its size in radians.

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    $\begingroup$ Angle is a vector $\endgroup$ – Oswald Feb 3 '16 at 9:51
  • $\begingroup$ imho: the reason why you normally dont see this, is simple: because it sucks.. Rotations with more than one angle needed (arbitrary rotations around arbitrary axes) are notoriously are source for brainf* And you have to be very careful how your geometry works (see e.g. euler angles). In a classical mechanics course you might cover spinning tops. However seeing that the angular velocity $\vec{\omega}$ is commonly used it is obvious that its time integral of course is also vectorial. $\endgroup$ – Bort Feb 3 '16 at 9:55
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    $\begingroup$ You are right, this is exactly what we do in practice. Be aware that we we do not define things such as angular momentum as vector because it is somewhat convenient, it it the way they behave that imposes their vector nature. $\endgroup$ – Dimitri Feb 3 '16 at 9:55
  • $\begingroup$ @Bort you seem to have gotten the closest to what I am looking for, I will do some reading if I can find anything about integrating angular velocity $\endgroup$ – Sebastian Giles Feb 3 '16 at 16:41
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    $\begingroup$ That second link is exactly what I was looking for, this answer in particular: physics.stackexchange.com/a/288/101191 sorry for the duplicate question $\endgroup$ – Sebastian Giles Feb 3 '16 at 21:44
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Can an angle be defined as a vector?

It depends on what you mean by "vector". If by "vector" you only mean something that has a magnitude and a direction, then yes, the axis-angle representation qualifies as a "vector".

To a mathematician, a vector is something that is a member of a vector space. In this context, the axis-angle representation fails to qualify as a vector. One problem is that there are many ways to represent a null rotation (e.g., a 360 degree rotation about any axis); the null element has to be unique for a space to qualify as a vector space. Another problem is that rotations do not commute; the composition of two elements has to be commutative for a space to qualify as a vector space.

Another issue: The time derivative of the axis-angle representation is not the angular velocity. It has little physical meaning. (Similarly, the integral of the angular velocity vector has little physical meaning.)

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Describing a rotation as a vector, with the direction of the vector along the axis of rotation, and the magnitude of the vector as the angle, is known as the axis–angle representation.

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  • $\begingroup$ According to the David's answer this does not satisfy the requested physical significance $\endgroup$ – Sebastian Giles Feb 3 '16 at 16:38

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