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Adding vectors is commutative because adding coordinates is commutative and vector addition is merely two applications of that same law vectors $A$, $B$

$$A + B = (\overrightarrow{a_1,b_1}) + (\overrightarrow{a_2, b_2}) = (a_1 + b_1, a_2 + b_2) + (c_1 + d_1, c_2 + d_2)$$

which implies Commutativity be default due to the fact that we are merely adding real numbers.

Edit: Mistook commutativity for associativity!

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  • $\begingroup$ I am seriously confused. What are $A,B$? What exactly are you trying to show? $\endgroup$ – BigbearZzz Dec 23 '18 at 4:51
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    $\begingroup$ associativity of vector addition operation -- axiom of vector space $\endgroup$ – Just do it Dec 23 '18 at 4:52
  • $\begingroup$ Just for context this problem is from "A Survey of Modern Algebra" - Garrett Birkoff, and Saunders Mac Lane, p.169. $\endgroup$ – PossiblyDakota Dec 23 '18 at 5:10
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    $\begingroup$ Not sure if it's not showing everything on my phone, but it seems like you have some c's and d's coning out of nowhere; you have to show $(A+B)+C=A+(B+C)$, so you should pick a side, start from it, and arrive at the other. $\endgroup$ – Ovi Dec 23 '18 at 5:16
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    $\begingroup$ @PossiblyDakota Commutativity of addition is also taken as an axiom... Can you quote the problem from "A Survey of Modern Algebra"? $\endgroup$ – Dair Dec 23 '18 at 5:23
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Your proof is confusing because $c_1, c_2, d_1$, and $d_2$ appear out of nowhere. But I think your general idea is correct. The proof is as follows.

The commutative property of two-dimensional real vectors is: For all two-dimensional real vectors $a = (a_1, a_2), b = (b_1, b_2)$, we must have $a + b = b + a$. This is true since

$$ a + b = (a_1 + b_1, a_2 + b_2) = (b_1 + a_1, b_2 + a_2) = b + a $$

where the second equality follows from the commutative property of real numbers and the other equalities follow from the definition of vector addition.

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