Product Sum for two rows from two dataframes and add sum to a new column

I have 2 df as Main df

main_df=pd.DataFrame(np.arange(3,9).reshape(-1,2), columns = list('AB'))

    A   B
0   3   4
1   5   6
2   7   8

weights df

w_df=pd.DataFrame(np.arange(1,5).reshape(-1,2), columns = ['wA','wB'])/10

    wA  wB
0   0.1 0.2
1   0.3 0.4

I need to multiply specific row from weights df into all columns of main df and put the summation into a column in main df so output should be like in case of using 1st row from weights df

Prod Sum of 1st row in main = 0.1 * 3 + 0.2 *4 = 1.1

Prod Sum of 2nd row in main = 0.1 * 5 + 0.2 *6 = 1.7

Prod Sum of 3rd row in main = 0.1 * 7 + 0.2 *8 = 2.3

i.e. output of main df should be like (in case of using 1st row from weights df)

    A   B   prod_sum
0   3   4   1.1
1   5   6   1.7
2   7   8   2.3