Why is vector addition defined the way it is?

[Mr Davis 97]

We all know that arithmetic addition arose not out of some axiomatic system, but out of the natural tendency to combine similar objects. I am satisfied with typical addition being loosely defined in this way. But when it comes to addition with other objects, like vectors, I am little bit confused. Why is that we define vector addition the way we do (i.e. a resultant vector is one that starts at the base of the first to the tip of the second)? Is this the definition of vector addition because it is the most "natural" way to "add" directed line-segments? Why does this definition happen to be so useful in physics?

 

[brainpushups]

Is this the definition of vector addition because it is the most "natural" way to "add" directed line-segments? Why does this definition happen to be so useful in physics?

The parallelogram addition of quantities like force have been known since antiquity (long before the concept of a vector was developed). Nature does not require that quantities add in this particular way, but it is really nice that they do! I would expect that the operation of vector addition is defined this way simply because the parallelogram addition of vectors has been experimentally verified.

 

[Geofleur]

Vectors are generalizations of ordinary arithmetic. All finite dimensional vector spaces are isomorphic to either $R^N$ or $C^N$, and to add vectors in either of these spaces, you add the components. This gives rise to the geometric construction for adding arrows, but the algebraic structure seems primary to me. Also, nature behaves linearly in so many instances, and when it doesn't, we find a way to linearize (usually). Scalar multiplication of vectors and vector addition are the essence of linearity. We break things up into pieces, operate on them with linear operators, and put the pieces back together again. Another addition rule, say a nonlinear one, would spoil our ability to do this.

 

[lightarrow]

There are two different concepts here:
1) Mathematics. Given the definition of vector as element of a finite dimensional vectorial space, it is isomorphic to (e.g.) R^n as geofleur wrote, so that geometrical rule for summing vectors in space is easily proved summing the components of two vectors.
2) Physics. The fact some quantities as velocity, force, etc, are vectors it's an experimental fact and it doesn't follow mathematically from something else (as long as I know).

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lightarrow

 

[sophiecentaur]

not out of some axiomatic system, but out of the natural tendency to combine similar objects.

When you compare the very familiar arithmetical operations with the slightly less familiar vector operations we perform every day, you may be reading more into things than necessary. The formal processes with vectors may be a lot more complicated than simple arithmetical processes but we are constantly using vectors on an informal basis in our lives. The guy with a boat who's dong a 'ferry glide' doesn't do the sums but he can bring his boat perfectly to a stop at the jetty on the river by applying them. The darts player is allowing for the drop due to g, in every throw.
The original rules for arithmetic were not based on axioms any more than the rules used for using vectors. Mathematicians got hold of those rules and, with a touch of Post Hoc reasoning, commandeered them and turned them into an art form. Still, they gave us back more than they took so we can hardly complain.
What exactly is Mathematics? How basic is it to the operation of the Universe?