I have been pondering the relationship between the nature of force as a vector quantity and the triangle law of addition. My question is:
Is force considered a vector quantity because it follows the triangle law of addition?
Or, does force follow the triangle law of addition because it is inherently a vector quantity?
I would appreciate it if someone could clarify this concept.
Force is considered a vector quantity because it has both magnitude and direction, and it obeys the fundamental rules of vector algebra, such as the triangle law of addition. The triangle law of addition is not the reason force is a vector; rather, force follows this law because it is inherently a vector.
The triangle law can define a possible sum of oriented segments. If we use such a definition of sum and define the multiplication by a scalar as a rescaling of the length of the oriented segment by that scalar (changing the orientation if the scalar is negative), we can easily prove that oriented segments with a common tail, equipped with such a sum and scalar multiplication satisfy all the axioms defining a vector space. I.e., oriented segments are vectors provided the two operations of sum and scalar multiplication are described this way.
Forces are not oriented segments, strictly speaking. However, if we can prove that we can establish a one-to-one correspondence between forces and oriented segments in D-dimensions, the composition of forces and the sum of oriented segments, and the rescaling of forces and rescaling of oriented segments, we can use such a mapping to show that forces, too, satisfy the vector space axioms.
On the other hand, if we know that forces are elements of a vector space of dimension D, we can use the isomorphism between all the finite-dimensional spaces of dimension D to represent the sum of two forces in terms of the triangle law.
Therefore, the two statements in the question are equivalent. Depending on the definition of force one uses, the other statement comes as a consequence.
Both can be correct, it depends on your point of view. Almost all people will tell you that force is a vector because it "behaves" like a vector.
Actually, when Newton formulated his laws, "vectors" did not exist, he described his laws with words and Euclidean Geometry, he knew that a force implies a change in the momentum, but he also noted that forces have a special behavior: the direction of the change of the movement was in the same direction of the force. Although vectors did not exist people knew intuitively what a direction is, in special they knew what means that a force has a direction.
Imagine you are in 1690 and you ask yourself "If I tie two ropes to a point on a box and I pull of them at the same time, in what direction the box will move? In the direction of any of the ropes? You will notice that the box goes towards you, not in a direction of any of the strings, although any force that you are applying to the box is not in direction towards you, so you have discovered something interesting, when a unique force is acting, the change of movement is in the direction of the force, but when there is more than one force this is not satisfied, the direction of the forces combine in some way to produce the direction of the change of movement.
If you continue doing similar experiments you will notice that the way this directions "combine" is the same as if you draw the forces direction at the same point, completes a parallelogram and the direction of the movement is the direction from one corner to the other. The concept of vector was influenced by the necesity of represent quantities that also has direction.
A physicist (that knows how forces behave) will say you "Forces behave like the vectors that mathematicians invented, in special they must follow the triangle law", but a mathematician (that knows the theory of vectors as abstract entities) will say you "Vectors behave like forces, of course the force is a vector, so vectors must satisfy the triangle rule"
Force is a vector (or, more precisely, can be modelled as a vector) because two or more forces can be added/combined using the triangle rule. This is an empirical observation. It is perfectly possible to imagine a world in which "forces" have both length and direction but are combined using something other than the triangle rule (e.g. the "sum" of two forces is whichever has the larger magnitude) - in which case these "forces" could not be modelled as vectors.
