  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 Land\'e 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.
