Pandas Python multiply column values by chosing coeffiients from row of another dataframe

Question

I can not figure out how to multiply a dataframe with the values selected as the row from another dataframe.

I have a number of factors that I observe for variables in a universe:

df_observations = pd.DataFrame([
    ['variable_1', 1, 1.1, 1.2], 
    ['variable_2', 2, 2.1, 2.2], 
    ['variable_3', 3, 3.1, 3.2] 
],
    columns=['observation',
             'factor_1',
             'factor_2',
             'factor_3'])

df_observations.set_index(['observation'], inplace=True)
df_observations

For each of the factors, I want to multiply by a coefficient, conditional on a state of the system

df_factor_state_coeff = pd.DataFrame([
    ['state_1', 10, 11, 12], 
    ['state_2', 20, 21, 22], 
    ['state_3', 30, 31, 32], 
    ['state_4', 40, 41, 42], 
],
    columns=['sys_state',
             'factor_1',
             'factor_2',
             'factor_3'])

df_factor_state_coeff.set_index(['sys_state'], inplace=True)
df_factor_state_coeff

I can only make a very "procedural" approach work:

for c in df_factor_state_coeff.loc['state_2'].index:
  df_observations[c] = df_observations[c] * df_factor_state_coeff.loc['state_2'][c]

df_observations

I feel like I am missing something here and should be able to do this as a multiplication of df_observations.multiply(df_factor_state_coeff.loc['state_2']) without having to loop but i cant figure that out. Would really appreciate a smarter way to do this.

thanks