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# Describing rotation in 3d with a vector

Learn how a three-dimensional vector can be used to describe three-dimensional rotation. This is important for understanding three-dimensional curl. Created by Grant Sanderson.

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• It feels like that this is not 3-D rotation, but 2-D rotation only. If you align the axis of rotation in z-direction, and then consider individual points, they are just moving in x or y direction. Very similar to something moving around a point in the x-y plane. If I am correct in visualizing this, can someone help me with an example where a point moves in all 3 directions, or is that not possible at all? As, when I think of it now, if I just consider one single point rotating around ANY axis, you can always align that axis to z-direction and then point will be moving in x-y plane only. I am little confused, please help
• Any momentary rotation is 2D in the sense that it must have an axis of rotation and everything rotates inside a plane that's perpendicular to that axis.
I think your confusion comes from thinking about "crazier" rotations but if you think about it all they really do is changing that axis of rotation with time, and that can also be described by allowing the curl to change and depend on t.
• At , What does Grant mean by putting vectors anywhere in space? Wont the rotation always take place around the axis of rotation (which is why the right hand rule works) which is essentially the vector arrow will be pointing?
• He's talking about the location of the arrow that represents the vector, as a whole... not about the tip of the arrow. You can place the arrow at the origin, or at the poles of the sphere, or anywhere, as long as the arrow has the same direction and magnitude, it is the same vector.
• When I think about a sphere rotating (like a tennis ball or something), I think about how it could be rotating about a certain axis, but then that axis itself could be rotation. Like maybe in 3-space it's rotating around the vector (1, 0, 0) in one instant, then around (0, 1, 0) the next, then around (-1, 0, 0), then around (0, -1, 0), then back to (1, 0, 0), etc. Will 3d curl describe this situation too?