I'm in Calc III right now, and I'm a little confused as to what constitutes "clockwise", and "counterclockwise" rotations when dealing with the various planes in 3d-space.
Of course, it's obvious in the 2d plane which direction of rotation is clockwise, and which is counterclockwise, but I'm unsure how this extends to the $xy, yz, xz$, or even arbitrary planes in $\mathbb{R}^3$.
Specifically, I'd like to know what the standard directions of rotation via the standard definition of the 3d rotation matrices (with their matrix-column vector products) are, and how/if that extends to rotations about arbitrary planes in $\mathbb{R}^3$.
I'm sure its something to do with the right hand rule, but It's not intuitively obvious to me what, and It's surprising to me that neither Wikipedia, nor my text, mentions this explicitly (that I can find).
Best Answer
Please correct me if I'm wrong, but from what it sounds like, you're looking for the Cartesian direction that a point will move in if rotated in some direction in $\mathbb{R}^3$. An infinitesimal rotation in $\mathbb{R}^2$ will move each point along it's tangent line. The direction of a counter-clockwise rotation is given by $(-y,x)$, and the direction of the clockwise rotation is $(y,-x)$.
In $\mathbb{R}^3$, it's almost just as easy to define the directions. The cross product of the rotation axis $\vec{n}$ by the point $\vec{p}$ gives the counter-clockwise tangent direction, and $\vec{p}\times\vec{n}$ gives the clockwise direction. In a left-handed system, these directions are flipped.
For a more geometric visualization of this, 3d rotations are just 2d rotations in the plane perpendicular to the rotation axis. In a right-handed system, counter-clockwise and clockwise directions are the same as they are in 2d.