1. Excuse my thickheadedness, but I don't get it.

SO(2) is a single parameter group. A vector transfromed to any other vector, where length is preserved with a single angle as parameter.

SO(3) is a three parameter group. If I take a vector in three space, it can be rotated in any direction with two angles.

Where does the third parameter come from?

2.

3. This should really be in the math forum, not the physics forum, but anyway...

SO(2) is the group of rotations in 2 dimensions. This can be specified with one angle.

SO(3) is the group of rotations in 3 dimensions. This can be specified with three angles. If you think of a sphere, you can specify any point with only two angles. The third comes from twisting around that point. (Stick your arm out in a direction. That's the first two angles. Twist your arm in place. That's the third.)

4. Originally Posted by MagiMaster
This should really be in the math forum, not the physics forum, but anyway...
The physics forum seems to need some uplifting, if you know what I mean.

SO(2) is the group of rotations in 2 dimensions. This can be specified with one angle.

SO(3) is the group of rotations in 3 dimensions. This can be specified with three angles. If you think of a sphere, you can specify any point with only two angles. The third comes from twisting around that point. (Stick your arm out in a direction. That's the first two angles. Twist your arm in place. That's the third.)
Thanks for responding Magi! But I still don't get it. A vector doesn't change rotated about its own axis. Is the idea that the inner product is invariant??

Edit: On second thought...

O(3) is defined such that

So a rotation of A about itself leaves |A| unchanged. This makes sense if the rotation of A about it's own direction is independent of phi and theta.

Have I got this right?

5. No, I don't know what you mean. This is still a math question and belongs in the math forum.

And, yes, rotating a vector about itself does nothing, until you start composing those rotations. But it's best not to think about rotating a single vector. Instead, think of rotating a set of orthogonal vectors.

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