Are angles in a Cartesian coordinate system vectors?

Question

For any given vector, denoting a point, (x,y), on a Cartesian coordinate system, if we consider the angle, phi, of this vector to the horizontal axes, x, is this angle, phi, considered a vector?

I would consider it a vector since it naturally provides a direction from the origin (used as a point of reference); however, it is confusing in that the angle itself has no direction, it is merely a magnitude, which grows or shrinks in 1-dimension, which can be observed in a polar coordinate system. Should it be a vector, then shouldn't it itself have a direction?

Answer 1

An angle, when observed on its own, is one dimensional & merely increases or decreases. This is better observable in the polar coordinate system, where it serves as an axes. As it only posseses magnitude, & no direction, an angle is a scalar quantity, & not a vector.

Answer 2

Vectors are not just directions, they must also have magnitudes. But, lets look at it from another, algebraic, perspective.

To become a vector, one needs to be able to add up with others and to be multiplied by a scalar. While adding angles is not a problem, multiplying by a scalar is somewhat problematic. Consider this: take $\alpha = 1$ (in radians) and multiply it by $2\pi$ to get $2\pi \alpha = 2\pi$ radians. But $2\pi = 0$ as an angle! In vector spaces we have a property that $\frac{1}{2\pi} 2\pi \alpha = \alpha$, but $\frac{1}{2\pi} (2\pi \alpha) = \frac{1}{2\pi} 0 = 0$. So, $1 = 0$, which is somewhat disappointing.

If you want a nice structure on angles, here it is: the circle group $\mathbb{T} \cong \mathbb{R} / 2 \pi \mathbb{Z}$ (the $2 \pi$ here is not that important: it is just the conventional measure of angles).

If, however, you are not dealing with angles, but with "changes of angles", those can be arbitrary real numbers! For example, if a solid rotates around some axis, it may happen that during some period of time the whole rotation is, say, $7\pi$. So, the angle changed by $\pi$, but the rotation that led to this angle was $7 \pi$. This way, angles form a group, known as real numbers $\mathbb{R}$, and also a vector space over $\mathbb{R}$, trivially.