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Polarized magnetic dipole (M1) emission lines provide important diagnostics for the magnetic field dominating the evolution of the solar corona. This paper advances a multi-line technique using specific combinations of M1 lines to infer the full vector magnetic field for regions of optically thin emission that can be localized along a given line of sight. Our analytical formalism is a generalization of the "single-point inversion" approach introduced by Plowman. We show that combinations of M1 transitions for which each is either a $J=1\rightarrow0$ transition or has equal Landé g-factors for the upper and lower levels contain degenerate spectropolarimetric information that prohibits the application of the single-point inversion technique. This may include the pair of infrared Fe XIII lines discussed by Plowman. We identify the Fe XIII 1074.7 nm and Si X 1430.1 nm lines as one alternative combination for implementing this technique. Our sensitivity analysis, based on coronal loop properties, suggests that for photon noise levels around $10^{-4}$ of the line intensity, which will be achievable with the National Science Foundation's Daniel K. Inouye Solar Telescope, magnetic fields with sufficient strength (${\sim}10$ G) and not severely inclined to the line-of-sight ($\lesssim 35^{\circ}$) can be recovered with this method. Degenerate solutions exist; though, we discuss how added constraints may help resolve them or reduce their number.