Negative Angles

Question

A lot of times text books refer to the measure of angles with a little arrow (see picture below).

But to go clockwise they represent it as below.

The thing is and maybe this is just me, but I think of this as a vector where it goes around the other way (so it would be just like the angle in the first picture).

$\textbf{Question:}$ Is there such a thing as angles represented in this way where they are assumed to act like vectors in space in this way? I just want more insight as to how to denote these angles correctly. For instance, could I denote an angle like below to denote a clockwise direction instead of the second picture?

Answer 1

" For instance, could I denote an angle like below to denote a clockwise direction instead of the second picture?", yes, you can, providing you show the angle of orientation. Such angles are called oriented angles.

Answer 2

In mathematics, $0^{\circ}$ is usually taken to point east and counterclockwise angles are usually taken to be positive (possibly because the mapping $x \mapsto \exp(2\pi ix)$ from the real line to the complex plane maps $0$ to $1$ and wraps the real line around the unit circle counterclockwise). In navigation, $0^{\circ}$ points north and clockwise angles (bearings) are taken to be positive (for reasons I am too uncertain to guess at).