Why is force a vector?
Uhm ... you start with an object at rest and notice that if you push on it in different directions it moves in different directions? Then notice that you can arrange more than two (three for planar geometries and four for full 3D geometries) non-colinear forces to cancel each other out (hopefully you did a force-table exercise in your class and have done this yourself).
The demonstration on an object already in motion is slightly less obvious but you can take the ideas here and generalize them.
In a sense this is so obvious that it's hard to answer because almost anything you do with forces makes use of their vector nature.
Vectors are things that add like little arrows. Arrows add tip to tail.
Number of rocks is not a vector. 2 rocks + 2 rocks = 4 rocks.
Displacement is a vector. If you move 2 feet left and 2 feet left again, you have moved 4 feet. Two arrows 2 feet long pointing left added tip to tail are equivalent to one arrow 4 feet long pointing left.
If you move 2 feet left and 2 feet right, you have moved back to the start. This is the same a not moving at all. You can't add rocks this way.
Force adds like this. Two small forces to the left are equivalent to a big force to the left. Equal forces left and right are equivalent to no force. This is why force is a vector.
Edit - The comments raise a point that I glossed over. This point is usually not raised when introducing vectors.
Mathematicians define a vector as things that behave like little arrows when added together and multiplied by scalars. Physicists add another requirement. Vectors must be invariant under coordinate system transformations.
A little arrow exists independently of how you look at it. A little arrow does not change when you turn so it is now facing forward. Equivalently, little arrows do not change if you rotate the arrow so that it faces forward.
This is because space is homogeneous and isotropic. There are no special places or directions in space that would change you or an arrow if moved to a new location or orientation. (If you move away from Earth gravity is different. If this matters, you must move Earth too.)
By contrast, a scalar is a single number that does not change under coordinate system transformations. Number of rocks is a scalar.
The coordinates that describe a vector change when the coordinate system is changed. The left component of a vector is not a scalar.
There is a 1-D mathematical vector space parallel to the left coordinate of a vector. If you rotate the coordinate system, it may be parallel to what has become the forward component. A physicist would not say it is a vector space.
A minor nitpick: force is not a vector. Like momentum, it is a covector or one-form, and covariant. You can see this in several ways:
- from the principle of virtual work: force is a linear function mapping infinitesimal displacements $\delta\mathbf{x}$ (a vector) to infinitesimal changes in energy $F\delta\mathbf{x}$ (a scalar) and hence a covector by definition.
- Newton's second law $F=ma$: acceleration is a vector, which is "index-lowered" by the mass to give force.
- conservative forces arise from the differential of potential energy, $F = -dV$, and the differential of a function is a one-form (covariant).
The difference between a vector and covector may not make sense if you're just starting to learn about physics, and for now, knowing that forces can be "added tip to tail" like vectors may be enough for practical calculations. But it is something you should start to pay attention to as your understanding matures: like dimensional analysis, carefully keeping track of what your physical objects are, mathematically, is helpful both for building deeper understanding, and catching errors.