A1 = np.array([1, 2, 3])
B1 = np.array([4, 5, 6])

dot_product = np.dot(A1, B1)
print("1. 点积 A*B =", dot_product)

def cosine_similarity(v1, v2):
    dot = np.dot(v1, v2)
    norm_v1 = np.linalg.norm(v1)
    norm_v2 = np.linalg.norm(v2)
    return dot / (norm_v1 * norm_v2)

cos_sim = cosine_similarity(A1, B1)
print(f"2. 余弦相似度 = {cos_sim:.6f}")
print(f"   精确分数形式: {dot_product}/({np.dot(A1, A1)}*{np.dot(B1, B1)}) = {dot_product}/{np.dot(A1, A1) * np.dot(B1, B1)}")

A2 = np.array([1, 0])
B2 = np.array([0, 1])
cos_sim_special = cosine_similarity(A2, B2)
print("3. 当 A = [1, 0], B = [0, 1] 时:")
print("   - 点积 =", np.dot(A2, B2))
print("   - 余弦相似度 =", cos_sim_special)