And how is the resultant formally calculated? Why do we treat force as a free vector when writing down Newton's Laws of motion?


As far as you are not concerned with rotations, you may treat forces as free vectors. Because, you know, translational motion is only determined by magnitudes and directions of applied forces.

However, when you want to investigate rotations, you have to treat forces as localized vectors, because what determines rotation is not force per se, but, rather, torque. And torque somewhat depends on point of application of force.

The reason I'm using the word "somewhat" is that you may still move forces around but in a very limited way: each force may be translated only along the line of that force.

As for the resultant, when you are concerned with rotations of an object, you calculate resultant treating forces like free vectors, but you have to localize resultant at the point, where lines of forces intersect (refer the figure, where $R$ is a resultant of $F_1$ and $F_2$). Your object will not rotate if and only if line of resultant force goes through the object's center of inertia.

An example


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