0

I have a reference frame $xyz$ and a vector $\mathbf{v}$ in this frame defined by the angles $\theta$ and $\varphi$ (the amplitude is not important). A rotation is performed, defined by the three Euler angles or the change in roll/pitch/yaw, which gives the frame $x'y'z'$ and the vector $\mathbf{v'}$.

Question 1: If I know only the two vectors $\mathbf{v}$ and $\mathbf{v'}$, can I determine the three angles describing the rotation performed?

From my readings (Is the rotation matrix unique for a given rotation?, $3D$ rotation matrix uniqueness, Expression of rotation matrix from two vectors), I understand that it is not possible, because there are infinite rotations that could change $\mathbf{v}$ to $\mathbf{v'}$. If you know a textbook or another source that demonstrates this I would be happy.

Question 2: Then, how many different vectors would I need to be able to determine the rotation performed? (e.g. if I have two vectors $\mathbf{v}$ and $\mathbf{w}$ that are changed to $\mathbf{v'}$ and $\mathbf{w'}$ and I know all of them, is it enough?)

illustration of the frame, vector, and rotation

James
  • 77
  • See @Hosam Hajjir's solution. The only thing I would add is that while you need two pairs of vectors, those vectors could come from the basis for the original frame and the basis of the rotating frame. The only reason I say that is because he used $v$ in his solution, and $v$ exists in your plot, but the two aren't necessarily the same. – mode_er Jan 30 '22 at 02:28
  • Yes, thanks for the clarification. – James Jan 30 '22 at 15:32

1 Answers1

1

You need two pairs of vectors $v, v'$ and $u, u'$ to determine the rotation matrix.

Because then, you can write,

$v' = R v $ and $ u' = R u $

In addition you can write, $ v' \times u' = R (v \times u) $

where $\times$ is the cross product. Hence, it follows that

$ [ v', u', v' \times u' ] = R [ v, u, v \times u ] $

and you'll be able to solve for the rotation matrix $R$ by matrix inversion,

$ R = [ v', u', v' \times u' ] [v, u, v \times u ]^{-1} $

Hosam Hajeer
  • 21,978
  • Yes, it makes sense, with two vectors and their cross-product, we have something similar to $xyz$, so we can get the rotation. Thanks for the clear explanation. – James Jan 30 '22 at 15:29