Can there be a new of adding vectors? Suppose we have a vector v=&lt;a,b&gt;. And another vector w=&lt;c,d&gt;. Now, when we do (v+w), we do &lt;a+c.b+d&gt;. I am asking why isn't there also another kind of vector addition such as (v+w)(We may use another appearance) =&lt;a+d,b+c&gt;?
I'm not saying there should be one. But I want some logics for it to not exist.
Reasons not accepted-
1.They are of no use in the real world-Mathematica has its own vast world. And I know there's many things in Mathematics that have no real counterpart...at least yet.
2.They don't have any geometrical interpretation-I don't think many ones have tried to give them geometrical meaning. It's like i before Gauss.
3.One dimension is independent of others- Give me reason why I can't add them above way I told.
Maybe they can open a new chapter of vector addition...
You can define vector spaces with very different kinds of addition, they only have to satisfy the vector space axioms. Therefore, when you choose a basis $\{v_i\}$ of your vector space, every vector needs to be able to be written as
$$ v = \sum_i \lambda_i v_i.$$
When we use component representation of a vector space we simply keep track of the $\lambda_i$, so we would write
$$ v = (\lambda_1,\dots,\lambda_n).$$
Therefore, if $w=(\mu_1,\dots,\mu_n)$, we have
$$v + w = \left(\sum_i \lambda_i v_i \right)+ \left(\sum_i \mu_i v_i \right)= \sum_i (\lambda_i + \mu_i) v_i = (\lambda_1+\mu_1,\dots, \lambda_n+\mu_n).$$
Thus the way that you add vectors in components is a direct consequence of the vector space axioms, and does not depend on the way we choose our addition in the vector space in the first place.
You can define vector spaces with very different kinds of addition, they only have to satisfy the vector space axioms. Therefore, when you choose a basis $\{v_i\}$ of your vector space, every vector needs to be able to be written as
$$ v = \sum_i \lambda_i v_i.$$
When we use component representation of a vector space we simply keep track of the $\lambda_i$, so we would write
$$ v = (\lambda_1,\dots,\lambda_n).$$
Therefore, if $w=(\mu_1,\dots,\mu_n)$, we have
$$v + w = \left(\sum_i \lambda_i v_i \right)+ \left(\sum_i \mu_i v_i \right)= \sum_i (\lambda_i + \mu_i) v_i = (\lambda_1+\mu_1,\dots, \lambda_n+\mu_n).$$
Thus the way that you add vectors in components is a direct consequence of the vector space axioms, and does not depend on the way we choose our addition in the vector space in the first place.