Three-Phase with Floating Neutrals

A three-phase AC circuit contains three ideal sinusoidal voltage sources and a load consisting of three resistors. The voltage sources share a common neutral, and the resistors share a common neutral. The two neutrals are floating, and are not directly connected to each other.

The voltage sources have values \((\vec{V_A},\vec{V_B},\vec{V_C}) = (1\angle0 ^\circ, \, 1\angle{-120} ^\circ, 1\angle120 ^\circ) \). The resistor values are shown in the diagram.

Consider the source neutral to be the voltage reference for the circuit \((V = 0)\). Let \(\vec{V_N}\) be the voltage at the load neutral with respect to the source neutral.

What is the magnitude of \(\vec{V_N}\), to 3 decimal places?

Note: All voltage magnitudes are given in RMS, and the answer is expected in the same format. The symbol "\(\angle\)" indicates the complex phase angle of a sinusoid.

Hint: The answer can be found by writing and solving one complex equation, equivalent to two real-valued equations.

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