polar.py

from math import acos, atan, cos, sin
from mathutils import Vector

def vect_to_polar(vect):
    x, y, z = vect
    
    r = vect.length
    phi = acos(z / r)
    theta = atan(y / x)
    
    return r, phi, theta


def polar_to_vect(polar):
    r, phi, theta = polar
    
    x = r * cos(theta) * sin(phi)
    y = r * sin(theta) * sin(phi)
    z = r * cos(phi)
    
    return Vector((x, y, z))


vect = Vector((0.2, 99.9, 12.7))
assert (vect - polar_to_vect(vect_to_polar(vect))).length_squared < 0.1

print(vect_to_polar(vect))