I feel like this is mostly an artifact of notation. The thing that is not allowed with addition or subtraction is simplifying to a single term; otherwise it is fine. Consider:
10x + 5y −5x −10y = 10x − 5x + 5y −10y = 5x − 5y
So, everyone reasons to themselves, what we have here is two numbers. But hark, with just a little more information, we can see more clearly we are looking at a two-dimensional number:
5x − 5y = 5
5x = 5y +5
5x − 5 = 5y
x − 1 = y
y = x − 1
Such as a line.
This is what is happening with vectors, and complex numbers, quarternions, etc.
I feel like this is mostly an artifact of notation. The thing that is not allowed with addition or subtraction is simplifying to a single term; otherwise it is fine. Consider:
10x + 5y −5x −10y = 10x − 5x + 5y −10y = 5x − 5y
So, everyone reasons to themselves, what we have here is two numbers. But hark, with just a little more information, we can see more clearly we are looking at a two-dimensional number:
5x − 5y = 5
5x = 5y +5
5x − 5 = 5y
x − 1 = y
y = x − 1
Such as a line.
This is what is happening with vectors, and complex numbers, quarternions, etc.