How can I get a 3D rotation matrix

How can I get a 3D rotation matrix by only knowing a degree? That means, firstly, set a degree, say, 10, then generate a 3D rotation matrix randomly in order that for any unit vector v from the origin, the angle between v and the rotated vector Rv would be 10 degree. Thank you so much!!

Answer 1

[Mike Garrity]

If you look at the command makehgtform , you'll find a form that looks like this:

M = makehgtform('axisrotate',[ax ay az],t)

The axis [ax ay az] needs to be normal to the vector you want to rotate. As others have noted in the comments, this matrix will apply the same rotation to all of the vectors which lie in the plane which is normal to [ax ay az], but it will rotate vectors which do not lie in that plane by a smaller amount.

The angle t is in radians, so you'll need to convert.

Also, this command returns a 4x4 matrix because it supports translations. You just want the upper 3x3.

Answer 2

[James Tursa]

You can't have a rotation matrix that rotates all vectors in 3D space such that the angle between the original and rotated vector is 10 degrees. You can get an R that will do this for all vectors in a specific plane, but not all vectors in 3D space. E.g., a vector along the axis of rotation will end up being rotated 0 degrees. Other vectors, in general, will be somewhere between 0 and 10 degrees.

[Liangke Zhao]

Thank you for response! There might be a misunderstanding. What I hope to rotate is only one vector at a time, not all vectors in the space. I'm sorry I didn't ask clearly.

[James Tursa]

Given a single vector v, you want a rotation matrix that rotates it 10 deg in a random direction?

[Walter Roberson]

You required that the rotation matrix rotate "any" unit vector from the origin by 10 degrees. That cannot work. There must always be an axis of rotation, and any vector along that axis will be left unchanged.

[Liangke Zhao]

Yes indeed. So I turned to choose an axis randomly. Thank you all!