The short version is we put a resistor in series with the coil. We need our generator at the frequency we expect to operate the circuit. We can adjust the resistor or the generator to get the desired voltage across the resistor. What is that value? It would seem one half the generator voltage would be dropped across the resistor. Because the circuit has reactance and the voltage is shifted in phase this is not the case. The resistor will drop one half the voltage as the phase shift is 45 degrees. At 45 degrees the voltage will be 1.414 times the supply which will put .707 across the resistor. You can see the resistor and coil are dropping 700mv with a 1volt signal. To do a more accurate measurement you would need a Maxwell Bridge.
The Maxwell Bridge uses adjustable resistors and inductors to null the bridge. The detector D will read a voltage either side of balance.
The vector analysis shows that E is the vector sum when E1 = E2 and E3 = E4. This would be the point where D is null.