# Sense of rotation. How would the rotation matrix look like for this “arbitrary” axis?

My first question is how do you define the sense of rotation about an arbitrary axis?

Rotations are usually counterclockwise and when referring to rotation with respect to the $x$,$y$ or $z$ axis they happen as you look towards the origin when standing on the positive side of the axis. E.g. if you were to rotate the xy plane by $\theta$ then the rotation would be counterclockwise ($\theta>0$) if you were looking at the xy plane from the positive branch of the $z$ axis.

But if I were to ask you to rotate your coordinate system around some arbitrary axis $y'$ how would you know the sense of this rotation?

The next part of my question is related to a homework problem, and is quite short.

I'm asked to have a rotation $\alpha$ in the xy plane. I've done so using the mentality above. Then I'm asked to rotate (in another instance, that is, not doing the rotations consecutively) my coordinate system by $\beta$ in the $x'$-$z'$ plane (where $x',y'$ and $z'$ were my axis after the first rotation).

I only know how to do this if I imagine the rotations done consecutively. But I think my professor is asking me to consider the rotations separately.

How is this done separately? How is the sense of rotation determined here?

My intuition is that it's the same as doing the rotations consecutively.

Also: Is "rotating the xy plane" the same as "having a rotation in the xy plane"?

Thanks.

• What do you mean by the "sense of this rotation"? What does "sense" mean here? – Ben Grossmann Jan 25 '15 at 18:51
• I mean this: For example, I have to rotate a system by $\theta$ "counterclockwise" around the z axis. There are two ways to achieve this (e.g. looking at the rotation process for one side, maybe "above", and looking at the process from "below") Both methods give a different "sense" of rotation. Is that a good definition of sense? So if my professor says in the "counterclockwise sense" then he should also specify the point of view. – DLV Jan 25 '15 at 18:56
• Okay, sure. The general rule is that the angle is counterclockwise when looking from the "positive direction" of the given axis. When they give you an axis $y'$ (presumably as a vector), there is an implied "positive" side. For example, rotating around $y' = (0,0,1)$ is the same as the usual rotation about the z-axis. – Ben Grossmann Jan 25 '15 at 18:59
• The phrasing in my homework is "obtain the rotation matrix equivalent to a rotation by $\alpha$ in the x-y plane" is this the same thing as "rotate by $\alpha$ the x-y plane"? Thanks. – DLV Jan 25 '15 at 19:13
• It's certainly the same as "give the matrix that rotates by $\alpha$ in the xy plane", if that makes more sense. – Ben Grossmann Jan 25 '15 at 19:18

Stick your thumb of your right hand along the direction of the rotation axis. A positive angle is along the way your fingers point.

Like this http://electron9.phys.utk.edu/Collisions/rotational_motiondetails.htm

• Does my thumb point in the positive direction of the axis? – DLV Jan 25 '15 at 21:10
• Yes. See the arrow in the picture showing the direction of the rotation axis. – John Alexiou Jan 25 '15 at 21:10

In 3D, the "sense of rotation" is not + or - but a vector (the axis of rotation), and opposite "senses" of rotation along this vector just mean reversing the axis (or making the angle negative, whichever).

The rotation matrix about an arbitrary axis can be obtained from Rodrigues' formula.