how did they arrive to the rule of addition of vectors?

Question

I'm an A2 igsce math student and I'm taking mechanics for the first time in math this session.

I wanted to ask about how they arrived to the rule of addition of vectors. How did they know that if we add the X's and Y's of two vectors they would get a third vector which has exactly the same direction and magnitude of the force that could replace these two vectors or forces.

I'm convinced that it's correct and can feel or see the direction a point will accelerate in if two certain forces are applied to it. And can feel how the resultant force tends to get closer to the bigger force and how if two equal forces are applied with the same angle the resultant force is going to be exactly between them.

So I'm sure who invented the vectors had the same feelings and visions too but how did he arrive at this simple method to get such fascinating and exact results, not only did he manage to get the direction but the magnitude of the resultant too!

Answer 1

Vectors don't exist in reality, they are just a mathematical model that works to describe some aspects of reality. Physics is not mathematics and not all propositions can be derived simply by logic(even if this fact seems to be true also in maths in a certain sense). So I think that many experiments have been made to grasp the nature of forces and how they "sum" to each other. You seem to be attached to an extremely linear model of history of mathematics and physics:there isn't "who invented vectors". Human mind is not a formal system and the thought that the mathematical and physical history follow the linear path of a modern book is pure illusion.

Answer 2

I think it is important to realize that reality is special. There is some structure to it, whether we can figure out what it is or not. And there is even radically different good approximations at different scales. At the human scale, objects in space seem to obey euclidean geometry. What does this mean? Well, for one thing, rigid motions (combinations of translations and rotations) preserve lengths and angles! Have you ever wondered if a solid object can be moved in some manner that its shape is changed? It just does not. Under any rigid motion $f$ in euclidean geometry, $|f(P)-f(Q)| = |P-Q|$ for every points $P,Q$, and $∠PQR = ∠f(P)f(Q)f(R)$ for every points $P,Q,R$. The fact that objects at the human scale seem to obey this tells us that euclidean geometry is a good approximation of the geometry of reality at the human scale (not too big or small). In fact, not just euclidean geometry by specifically the euclidean space $ℝ^3$ seems to be a good approximation for human-scale geometry of reality. And $ℝ^3$ is a vector space and obeys vector addition in the sense that you noted.

In other words, we would not be so interested in $ℝ^3$ and vector spaces in general if reality did not in the first place have structure that looks like $ℝ^3$ at a scale we can observe! Similarly, you could ask why people came up with the axiomatization of the naturals as a discrete ordered semi-ring plus induction, and the answer is again the same! Namely, counting seems to obey these properties!

As for the addition of forces specifically, it stems from the relation between force and acceleration in newtonian mechanics (which is at the human-scale). We have that a force $F$ on a mass $m$ induces acceleration $a$ given by $F = m·a$, and $a$ is the second derivative of position $x$. Now a position is a vector in $ℝ^3$ (according to our good approximation), so acceleration is also a vector in $ℝ^3$, and adding two forces is observed to induce the sum of the corresponding accelerations, and hence two forces applied on the same mass induce exactly the same acceleration as the vector sum of those forces applied on that mass. Take a while to think through this; it is indeed not merely an issue of discovery, but there is some underlying mathematical constraint given the observed relation between forces and positions via acceleration.