# Rotation orientations in n-dimensions

I'm doing a change of variables that involves doing simple rotations on the standard basis vectors in R^n, and I'm wondering what the standard orientations are in n dimensions are and why. For example, in 3d, looking down the +z axis, +y is CCW from +x and down the +y axis, +x is CCW from +z. This seems obvious, but in 4 dimensions, if we assume that x,y and z have the standard orientations when looking down the +u axis, then when looking down the +z+y plane, is +u CCW from +x or the other way round? And in 6d? The simple rotation matrices in 4d here:

http://ken-soft.com/2009/01/08/graph4d-rotation4d-project-to-2d/

don't seem to have any discernable pattern to me...

I suppose, specifically, I need to know how to methodically create simple rotation matrices in n-dimensions, but if anyone knows where I can learn something deeply about the subject I would appreciate a reference.

## migrated from mathoverflow.netApr 11 '14 at 5:34

This question came from our site for professional mathematicians.

Let me suggest one article, which explicitly describes the simple rotation in $\mathbb{R}^n$ as rotation about an $\mathbb{R}^{n-2}$ subspace:

Mortari, Daniele. "On the Rigid Rotation Conept in $n$-Dimensional Spaces." Journal of the Astronautical Sciences 49.3 (2001): 401-420. (PDF download link)

This reaches the following expression as the $n \times n$ orthogonal matrix that represents a rigid rotation in $\mathbb{R}^n$:

where $J_2 = \left[ \begin{array}{cc} 0&-1\\ 1&0 \end{array} \right]$, and the meaning of the remaining notation can be found in the paper. Here is a neat figure later in the same paper:

By bending the axes, Figure 2 artistically provides a "way to see" the geometry of coning about the subspace identifed by the three Orthogonal axes $a_1$, $a_2$, and $a_3$.

• Awesome. Also, for anyone else reading this, P is the $n\times 2$ matrix describing the plane of rotation, so, for example, rotating in such a way that a positive angle takes the positive $x$ axis to the positive $y$ axis, $P$ would be $[X,Y]$, where $X = [1,0,0,0..]^T$ and $Y = [0,1,0,0...]^T$ – Scott Apr 11 '14 at 14:16