Recall that vector addition can be visually represented by the head-to-tail method. This means that the head of one vector is glued to the tail of the other vector, to point us directly to the target vector we're after. Think of it like following instructions on a map.
A visual example is shown below. There are two pathways to arrive at the c = a+b vector. The patheway on the right is a+b while the path on the left is b+a. The order doesn't matter since we arrive at the same destination.
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Here's an algebraic approach without the need of a drawing.
Consider that vectors a and b are defined like so
a = <p,q>
b = <r,s>
where p,q,r,s are real numbers. They define the coordinates of the two vectors.
Adding vectors a and b gets us
a+b = <p,q>+<r,s> = <p+r,q+s>
We add the corresponding coordinates. Since p+r is the same as r+p, we can say,
b+a = <r,s>+<p,q> = <r+p,s+q> = <p+r,s+q> = a+b
Therefore, a+b = b+a. Again, this all relies on the fact that addition of scalar numbers can be done in any order. So by extension, vector addition can be done in any order. This applies to multiple vectors and not just two of them.
