In statics, we learn that a force applied to a rigid body is determined by its line of action and its magnitude. In case two forces are acting, we can construct the parallelogram of forces (by building a parallelogram on their lines of actions and finding its diagonal) to find the "resultant force" acting on the body.
When the two forces act along parallel lines, we can't build a parallelogram directly, but adding "nothing" to the system in the sense of two opposed equal forces and pairing them off with the original ones we can reduce the problem to a non-parallel case. Except, that is, when the original forces are parallel and opposite and equal in magnitude. Then we have a "couple" and not a force.
Is there a mathematically elegant way to understand this, that doesn't treat "parallel lines of action" as an exception to the general rule, and "couple" as an exception to an exception? Perhaps there's a general formula for the resultant force (given original lines of action and magnitudes) that "just works" for the parallel case, and gives results for the couple case that happen to lie in some suitable extension of the plane (perhaps the projective plane...?). Is there something like that, or another way to unify these cases?
A force $\vec{F}$ can be generally expressed in terms of its normal components, i.e.
$$\vec{F} = x\vec{i}+y\vec{j}+z\vec{k}$$
where $(x,y,z)$ is the components of the force in the Cartesian coordinates. The resultant force from two forces $\vec{F_1}$ and $\vec{F_2}$
$$\vec{F_1} = x_1\vec{i}+y_1\vec{j}+z_1\vec{k}$$ $$\vec{F_2} = x_2\vec{i}+y_2\vec{j}+z_2\vec{k}$$
is given by,
$$\vec{F_S} = (x_1+x_2)\vec{i}+(y_1+y_2)\vec{j}+(z_1+z_2)\vec{k}$$
The above formula is general, applicable to any two forces, including two parallel ones. For example, the resultant force of the two parallel forces $\vec{F_1}=a_1\vec{i}$ and $\vec{F_2}=a_2\vec{i}$ is $\vec{F_P} = (a_1+a_2)\vec{i}$. If they are opposite and equal magnitude, i.e. $a_1=-a_2$, $\vec{F_P}$ is simply becomes zero.
In the general force representation above, the parallel case is not an exception, rather it is encompassed as a special case.