import math

A = [1, 2, 3]
B = [4, 5, 6]

dot_product = sum(a * b for a, b in zip(A, B))
print("点积 A · B：", dot_product)

norm_A = math.sqrt(sum(a**2 for a in A))
norm_B = math.sqrt(sum(b**2 for b in B))
cos_similarity = dot_product / (norm_A * norm_B)
print("余弦相似度：", cos_similarity)

A_special = [1, 0]
B_special = [0, 1]
dot_special = sum(a * b for a, b in zip(A_special, B_special))
norm_A_special = math.sqrt(sum(a**2 for a in A_special))
norm_B_special = math.sqrt(sum(b**2 for b in B_special))
cos_special = dot_special / (norm_A_special * norm_B_special)
print("\n特殊情况的余弦相似度：", cos_special)
print("原因：两个向量相互垂直（正交），点积为 0，方向完全不同，因此余弦相似度为 0")