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Lets take two points, $(x_1,y_1)$ and $(x_2,y_2)$ say, $(1,0)$ and $(2,3)$ - We would write the distance formula as $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ with reference to cartesian plane. These ordered pairs with reference to x and y axes are also called as cartesian coordinates.

Now, for calculating distance in multiple dimensions, we would translate this formula generically as $\sqrt{\sum_{i=1}^n(X_i-Y_i)^2}$

If we apply the second formula to our example, the given points can be seen as $X=(1,0)$ and $Y=(2,3)$ where $X_1=1$,$X_2=0$ and $Y_1=2$, $Y_2=3$.

Please help me with the below two clarifications.

  1. The terminology for representing points in the second approach that is different from cartesian coordinate system. Are these what we call as axioms? Is this a way of representing points in Euclidean plane and not in cartesian plane?

  2. Is Euclidean plane different from cartesian plane as in it doesn't have reference axes for representation?

Tyberius
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Naseem Ahamed
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  • Does this answer your question? What is the difference between Euclidean and Cartesian spaces? –  Jun 16 '21 at 18:06
  • Using $X$ and $Y$ to represent two points is not a good choice, in my opinion, because it can cause some confusion. Why don't you call them $A$ and $B$? In this case: $A_1=x(A)$, $A_2=y(A)$, and so on. Using indices is necessary when there are many coordinates, but there is no real difference: these are cartesian coordinates in an arbitrary number of dimensions. – Intelligenti pauca Jun 16 '21 at 21:09
  • @NaturalNumberGuy That thread gave me a few leads before I posted this question and yet, I am not sure whether I am going in the right direction. In doubt, I suspected whether they would also come under the umbrella of metric spaces and normed vector spaces to understand such a system. – Naseem Ahamed Jun 17 '21 at 05:53
  • @Intelligentipauca "these are cartesian coordinates in an arbitrary number of dimensions." - Glad to see this obvious point here. So, there is no need for me to give a doubt of whether it belongs to a different system of representation? If it is the answer, can you please post it as an answer instead? – Naseem Ahamed Jun 17 '21 at 06:02

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Cartesian plane means Euclidean plane where two orthogonal (i.e. perpendicular) lines are given, called cartesian axes, and on each of them a unit is given, so that every point of an axis is labelled with a unique real number. (Non-perpendicular axes are also possible, but I'm not going to explain that here).

This setting allows to label any point $A$ of the plane with a unique pair of real numbers, called its coordinates; for instance: $A=(3,-5)$. We traditionally refer to the first coordinate as $x$ and to the second one as $y$, but that is not a rule. They could also be named $(h,k)$, or $(x_1,x_2)$, or in any other way, if we find that convenient.

The same structure can be set up in three-dimensional space, with three axes (pairwise orthogonal) and three coordinates for any point, which are traditionally labelled $(x,y,z)$. But it is often useful to define abstract spaces having $n$ dimensions, endowed with $n$ orthogonal axes and $n$ coordinates. In this case it is not convenient to label the coordinates as $(x,y,z,...)$ and one usually employs subscripts: $(x_1,x_2,\ldots,x_n)$. And for some point $A$ in that space we could for instance write $A=(A_1,A_2,\ldots,A_n)$.

As before, those notations are just a matter of convenience; different authors can use different names.

Intelligenti pauca
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