# Resultant of two forces acting in the same line

I'm quoting the definition of Resultant of two forces acting in the same line from the book "A FIRST COURSE IN PHYSICS" one of whose authors is Robert Andrews Millikan:

The resultant of two forces is defined as that single force which will produce the same effect upon a body as is produced by the joint action of the two forces.

I'm really confused as to whether the resultant of two forces say $A$ and $B$ is the force which is produced as a result of the two forces just mentioned or is it a completely separate force which is not caused by $A$ and $B$, but its effect is the same as that of the force produced as a result of $A$ and $B$? Even though, the force caused by $A$ and $B$, let's say it is $C$, is equal in magnitude to that of the resultant $R$ of $A$ and $B$ and the direction is also the same, but $C$ is caused by $A$ and $B$; however, $R$ has no primary causes as $C$ has. This is what I conclude from this definition; however, I'm not sure yet.

Mathematically, given two forces $\mathbf F_A$ and $\mathbf F_B$, the resultant is simply their vector sum; \begin{align} \mathbf F_A + \mathbf F_B. \end{align} This is consistent with the definition in your quote because it is a physical fact that if these two forces both act on an object of mass $m$, then the acceleration of the object will satisfy \begin{align} \mathbf a = \frac{\mathbf F_A + \mathbf F_B}{m}. \end{align} But here's the interesting thing. Let's say, for example, that the two forces $\mathbf F_A$ and $\mathbf F_B$ are produced by two people pushing on a rigid box. If, instead, a third person were to push on the box with a force $\mathbf F_C$ that is equal to their resultant \begin{align} \mathbf F_C = \mathbf F_A + \mathbf F_B \end{align} then the resulting motion of the box would be exactly the same. In other words, the motion of the box is insensitive to precisely what makes up the total force on it, all that matters is what the total force vector is.
The word resultant implies that this $R$ is determined by some prior force(s) (or primary causes as you put it) such as $A$ and $B$. I think your question is likely coming from the use of the word resultant, which implies the existence of primary causes on top of the fact that it already implies a mathematical equivalence.
E.g. $F=(+4-6)\hat x$ is completely equivalent to $F=-2 \hat x$
i.e, any problem requiring the use of the $F$ in the x axis would be completely and just as accurately be solved using either--the solution doesn't care. The equivalence implication of the word resultant is as axiomatic as saying 4-6=-2. The reason why the word resultant is used instead of equivalent is to show that the simple equivalent (such as $R$) has primary causes (such as $A$ and $B$), thereby making it valid to use this simplification in the first place ($R=C$ because $A+B=C$). Because ultimately, using resultant values makes no difference to the solution, it just makes the problem look a lot easier, but the step remains to show the calculations of how you arrived at your simplified/resultant/equivalent input value.