Composite Load Voltage

The load voltage is the sum of the scaled input voltages. As long as the magnitude of the load voltage is 10, it doesn't matter what the angle is.

$2 \times 3 \angle 0^\circ + 3 \times 4 \angle 60^\circ + 4 \times \vec{V} = 10 \angle \theta$

Consolidate the known terms:

$15.8745 \angle 40.8934^\circ + 4 \times \vec{V} = 10 \angle \theta$

We can get the smallest $\vec{V}$ if we choose a load voltage angle equal to 40.8934 degrees.

$15.8745 \angle 40.8934^\circ + 4 \times \vec{V} = 10 \angle 40.8934^\circ \\ \implies \vec{V} = 1.4686 \angle -139.1066^\circ$