Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Rotations in 4 dimensions are performed around a fixed plane, they can be described by $SO(4)$, which is a group of orthogonal matrices with determinant equal to 1. It is easy to derive rotation matrices around the coordinate planes in $\mathbb{R}^4$, for example, $$ \begin{pmatrix} \cos(\theta) & \sin(\theta) & 0 & 0 \\ -\sin(\theta) & \cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$ performs a rotation around $X_3X_4$-plane by an angle $\theta$. What I am interested is are rotations around an arbitrary plane in $\mathbb{R}^4$. Given such a plane, how to get a rotation matrix for this plane?

share|cite|improve this question
I notice you were careful to put "a rotation matrix." I really have to ask: was it intentional because you knew there is really more than one such matrix? :) – rschwieb Mar 18 '13 at 20:43
Thanks for you answer. I knew that there is more than rotation matrix. – Jimmy R Mar 18 '13 at 21:14
up vote 3 down vote accepted

Let $v,w$ be an arbitrary orthonormal pair of vectors in that plane. Extend it to an orthonormal basis of $\Bbb R^4$. Then the matrix that you gave above is a rotation matrix in terms of that basis.

To express it in another basis, just perform a change of basis to your desired basis.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.