When I think of a space-time vector, I think of it as a geometrical object in Minkowski space with the physical properties of magnitude and orientation wrt other space-time vectors. I can use four numbers to represent it uniquely using measurements from rulers and clocks that are physically different to one another. From this, I can define the differential space-time vector as:
$$ dx_i = (d\vec r, dt)$$
Instead, physicists decide to convert the time component of the above into a space interval measurement through multiplication by $c$, giving the differential four-vector as:
$$ dx_i = (d\vec r, cdt)$$
To me, it now looks like a vector with four space components which isn't consistent with the original space-time vector where the time component is physically different, and hence can't be combined with the other three space components: Why is this conversion done?