How to define distance for vector of angles? I have a vector of angles and I am looking for a method to compute the distance of my vector with any other vector of angles? I am looking for something similar to Euclidean distance but I know that Euclidean is for non-circular values.
I should mention that I want to use this function as a cost function in an optimization process to find a vector of angles that is closest to my own vector of angles. How can I define something similar to the Euclidean norm for angular (circular) values?

You are right that circular data requires special metrics. The distance between 1° and 359° should be small, e.g., not large as $|359-1|$ would suggest. And a circular metric must be "rotation-equivariant", i.e., it must not depend on the arbitrary choice where the angle 0° is located.
For distance measures, the cosine distance is often used and simple to compute:
$$d(\varphi,\vartheta)=1-\cos(\varphi-\vartheta)$$
You can generalize it to vectors by component-wise addition.
It is also possible to define indices somehow representing mean values and variances for circular data. See

Jammalamadaka, SenGupta: "Topics in Circular Statistics." World Scientific, 2001

