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If I rotate an object about an arbitrary axis, I can draw an arrow along that axis, and assign it a length proportional to the amount of rotation. I can project that arrow onto the basis of a rectangular coordinate system. The question arises, is it a vector?

I can certainly transform it from one coordinate system to another, preserving its meaning. I could add them together using standard vector addition, and assert that this sum means to rotate about the resulting axial by the resulting magnitude.

What doesn't work is to perform the rotation determined by one such arrow, and then perform the rotation for a second arrow and, in general get the rotation resulting from the vector sum of these.

http://sphotos-a.xx.fbcdn.net/hphotos-ash3/75130_484117724951803_2017520621_n.jpg [Broken]

Now, with displacements, for example, I can equate subsequent displacements and the summation of vectors.

Now, if I require that the combination of the rotation arrows be done simultaneously, using some real number parameter, such that both angles about the respective axes of rotation increase in proportion to the magnitudes, then, I believe, the result will be the same as the rotation defined by the vector sum.

The reason I got to thinking about this is because I was pondering Feynman's unproved assertion that angular velocity vectors can be summed.

Is this something that has an intuitive discussion in textbooks? Is there a position axial vector analogous to the polar position vector used to depict a displacement from an origin?

I can certainly transform it from one coordinate system to another, preserving its meaning. I could add them together using standard vector addition, and assert that this sum means to rotate about the resulting axial by the resulting magnitude.

What doesn't work is to perform the rotation determined by one such arrow, and then perform the rotation for a second arrow and, in general get the rotation resulting from the vector sum of these.

http://sphotos-a.xx.fbcdn.net/hphotos-ash3/75130_484117724951803_2017520621_n.jpg [Broken]

Now, with displacements, for example, I can equate subsequent displacements and the summation of vectors.

Now, if I require that the combination of the rotation arrows be done simultaneously, using some real number parameter, such that both angles about the respective axes of rotation increase in proportion to the magnitudes, then, I believe, the result will be the same as the rotation defined by the vector sum.

The reason I got to thinking about this is because I was pondering Feynman's unproved assertion that angular velocity vectors can be summed.

Is this something that has an intuitive discussion in textbooks? Is there a position axial vector analogous to the polar position vector used to depict a displacement from an origin?

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