Find the resultant force vector F_R.

To find the magnitude of the resultant force, we first need to find the resultant force vector by adding the two given force vectors. Then, we calculate the magnitude of this resultant vector. The phrase "along a horizontal floor" implies that we should consider only the components of the force that are parallel to the floor (i.e., the $\vec{i}$ and $\vec{j}$ components) for the effective force causing motion along the floor.

Step 1: Find the resultant force vector $\vec{F_R}$.
The two forces are given as:

$$\vec{F_1} = 2\vec{i} - 4\vec{j} + \vec{k}$$ $$$$