Examples
Multiply a DataFrame with a Series:

import numpy as np
import pandas as pd

df = pd.DataFrame([[0, 2, -1, -2], [2, 2, 2, 2]])
s = pd.Series([2, 1, 2, 1])
df.dot(s)

0    -2
1    12
dtype: int64

Multiply a DataFrame with another DataFrame:

other = pd.DataFrame([[0, 2], [1, 2], [-2, -1], [2, 0]])
df.dot(other) 

	0	1
0	0	5
1	2	6

The dot method give the same result as @:

df @ other

	0	1
0	0	5
1	2	6

The dot method works also if other is an np.array:

arr = np.array([[0, 2], [1, 2], [-2, -1], [2, 0]])
df.dot(arr) 

	0	1
0	0	5
1	2	6

Shuffling of the objects does not change the result:

s2 = s.reindex([2, 0, 1, 3])
df.dot(s2)

0    -2
1    12
dtype: int64