from math import *

# Data type of v is a python list or tuple or whatever
def magnitude(v):
    return sqrt(sum([c*c for c in v]))

def dot_geo(v1, v2, theta):
    return magnitude(v1) * magnitude(v2) * cos(theta)

def dot_alg(v1, v2):
    return sum([c1 * c2 for c1, c2 in zip(v1, v2)])

def cos_from_vectors(v1, v2):
    return dot_alg(normalize(v1), normalize(v2))

def normalize(v):
    m = magnitude(v)
    return [c/m for c in v]