Do vectors imply the superposition principle?

Question

I don't understand why it is necessary to state the superposition principle. Reading Griffiths' Introduction to electrodynamics, the superposition principle is described as experimental fact that is not implied by Coulomb's law. Well, it is clear to me that non-linear phenomena can exist, but as far as know, Couloumb's law for two point charges is:

\begin{equation} \vec{F}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{\left|\vec{r_1}-\vec{r_2}\right|^3}\left(\vec{r_1}-\vec{r_2}\right) \end{equation}

with obvious notation. Now, defining it as a vector implies that is sums like vectors ans thus linearity, so this should imply the principle of superposition.

Things would be different if we defined only the modulus of Coulomb's force (without saying it is a vector) and in that case, superposition would be implied.

So, at this point my guess is that first we observe that superposition principle is valid and for this reason we use vectors for forces.

What am I missing here?

P.S. I used electrostatics as a mere example here. The doubt is about the relation between vectors and superposition.

Answer 1

Superposition principle and using vectors for modeling forces are entirely independent things.

Using vectors for modeling forces

In Classical Mechanics, forces either are defined through the Second Law ($m \mathbf{a} = \mathbf{F}$), or are introduced as primitive entities which satisfy all the axioms of a vector space over $\mathbb{R}$. In all cases, the properties of a vector are built-in in the classical force concept.

Force superposition principle

It says something about the internal forces in the case of a system made by more than two bodies. In particular, it says that the force on body n.1, due to the simultaneous presence of a body n.2 and a body n.3, is just the vector sum of the forces that body n.2 would exert without the presence of body n.3, and similarly for the force exerted by body n.3. Formally, if $\mathbf{F}_{1i}$ is the force on body n.1 due to the presence of the body $i$-th body alone, the superposition principle says that the total force on body n.1, due to the presence of $n-1$ other bodies is $$ \mathbf{F}_1 = \sum_{i=1}^n \mathbf{F}_{1i} $$ In many cases, including the case of the Coulomb force, the superposition principle is strictly true. However, this is not always the case. Electrostatics provides cases where the superposition principle may not hold.

Let's consider the case of three static charged and polarizable point-like bodies. The dipole moment induced on body n.1 depends of the total electric field at its position, which is due to the charges on the other bodies, but also to the induced dipole moment on these bodies, which in turn depend also on the dipole moment on body n.1. This is a self-consistent problem that is equivalent to minimize the total electrostatic energy of the system with respect to the dipole moments.

Therefore, each induced dipole depends on the position of all the other bodies and, in general, the resulting force is not reducible to a sum of pair-wise forces.

Answer 2

You're always going to use vectors for forces, because forces have magnitude and direction, so nothing else really models them well. Even if the superposition principle was not valid we would still have to use vectors. We would have to have to use some law $\vec{F} = \vec{F}(\vec{r},Q_1, \ldots , Q_n)$.