Are positive and negative angles defined for three dimensions

Question

In two dimensions, the anticlockwise direction is considered to be positive and the clockwise direction is considered to be negative. Does this hold even for angles in 3 dimensions?

Answer 1

In 3 dimensions, you'll have to specify the axis of rotation. This means that you have 3 vectors involved: The two vectors $a$, $b$ for which you want to determine the angle, and the axis $r$ around which you are rotating.

Whether the rotation is clockwise or anti-clockwise is a matter of definition. For example, if you look at a clockwise rotation from above, then it will be anti-clockwise when you look at it from below. This is the reason for why you need a 3rd vector (or some other means) to determine clockwise-ness.

One way to determine whether a rotation is clockwise, could be to define clockwise-ness depending on whether

$$\operatorname{sign}(\det(a, b, r)) \tag1$$

is $1$, $-1$ or $0$. where $0$ basically means "undefined". The latter case occurs when one vector is 0, or when $a=\lambda b$, that is if the angle is $0^\circ$ or $180^\circ$. It also occurs if $r$ is in the plane spanned by $a$ and $b$.

Notice that $\det$ is alternating multi-linear, thus the sign of $(1)$ will flip when $a$, $b$, $r$ are permutated by an odd permitation. It also flips when one of the involved dectors is replaced by its negative, for example flipping $r$ will flip clockwise-ness: $$\operatorname{sign}(\det(a, b, -r)) = -\operatorname{sign}(\det(a, b, r)) $$ etc.

Also notice that one $r$ won't work for all $a$ and $b$. For example, if $a$, $b$ and $r$ are linerly dependent, then you'll get the "undefined" case. This means you are looking at the rotation edge-on.