I know my question seems extremely vague or meta, but I've been studying linear algebra since around the beginning of September and we've recently touched on the concept of Vector Spaces, which changed my idea of what vectors actually are.
I'm of a physics background, so to me, a vector is a dimensioned quantity with magnitude and direction, and the fact that dimensions like forces, velocities and torques have the properties of vector mathematics is basically axiomatic, at least that's how I've understood it as. So to me, they were a projection of a quantity with unique, orthogonal components. In my linear algebra course, I'm interpreting them more generally as any element of a vector space, which means elements of a set which satisfy the vector space axioms, which are extremely and surprisingly general. So, for example, any 3-tuple that satisfies something like, $2x + 4y+ 3z = 0$ (this is completely arbitrary - I don't know if this exact example even satisfies the axioms but bare with me for the sake of the point). So, if those 3-tuples are vectors, and so is $\vec F = 3 \hat i + 4xy \hat j$, I feel like I'm missing some key understanding as to what makes a vector a vector if so many things can be it.
For example, how exactly is a vector different from a point in coordinate space, other than one obviously being a vector and one being a coordinate, one having magnitude and direction, and one not, etc? Plus, consider, say, my point in coordinate space $(x,y,z)$ such that it satisfies the set condition I said previously that $2x+4y+3z=0$. Can I now draw a line to that point and call it a vector?
If anything about my question is confusing, please let me know and I'll try to elaborate.