# Is the line element a distance vector or displacement vector?

In my Electrodynamics and Electromagnetism course, the professor is deriving Maxwell equations and the electromagnetic field tensor from differential geometry and wants to show how special relativity is built in Maxwell equations. To be honest the mathematical steps aren't that hard to follow but rather the physical meaning behind the terms. for example in three dimensional Minkowski space the line element is:

$$ds^2 = -(cdt)^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2$$

you can either use transformation or simply use the chain rule to change coordinates, let's say for a arbitrary moving coordinates, we get:

$$ds^2 = g_{\mu \nu}\check{dx}^\mu\check{dx}^\nu$$

and lets say there is another stationary coordinates

$$ds^2 = -(cdt)^2$$

if we take 2 events, let's say the stationary coordinates moved through time for a period, then the distance in space-time (which is a vector) would be

$$ds^2 = -(cdt)^2$$

which is equal to its proper time (I would also love a an easy definition of proper time).

but then you could say that the moving coordinates agrees that the events happened and the distance between them is

$$g_{\mu \nu}\check{dx}^\mu\check{dx}^\nu$$

but then what is meaning of the last term? is it the measurement of the space-time distance the stationary coordinates moved with respect to this coordinates? if so then why would it be equal to the proper time of the stationary coordinates?

I don't think I formulated the question properly but this is kind of a new language to me. I appreciate the help, Thanks!