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$x,y$ are perpendicular if and only if $xcdot y=0$. Now, $||x+y||^2=(x+y)cdot (x+y)=(xcdot x)+(xcdot y)+(ycdot x)+(ycdot y)$. The middle two terms are zero if and only if $x,y$ are perpendiculat. So, $||x+y||^2=(xcdot x)+(ycdot y)=||x||^2+||y||^2$ if and only if $x,y$ are perpendicular. ( I copied this)

I think this argument is circular because this property :

$xcdot y=0 $ implies $x$ and $y$ are perpendicular

Comes from Pythagorean theorem.

Oh, it just came to mind that the property could be derived from law of cosines. Law of cosines can be proved without Pythagorean theorem right? ,so the proof isnt circular?

Another question: If the property comes from Pythagorean theorem or cosine law, then how does dot product give a conditon for orthgonality for higher dimensions ?

Remember that $mathbb{R}^n$ as a vector space over $mathbb{R}$ with standard addition and scalar multiplication isn't rich enough to do geometry. When we talk about geometry, particularly the Euclidean geometry, we want to study the transformations that keep some properties like length and angles preserved, namely isometries. Most of our theorems will be about how two angles, lengths or circles are congruent and two shapes are defined to be congruent if and only if they can be transformed into each other by one of these 'transformations' (isometries). Viewing geometry from this point of view, i.e. through its group of transformations, is related to what is called the Erlangen program.

Anyway, it turns out that both of these properties, i.e. length and angle, can be packaged into one algebraic concept called the inner product (which I'm going to define below). More precisely, if $vec{x} in mathbb{R}^n$, then

$$| vec{x} | = sqrt{langle vec{x}, vec{x} rangle}$$
$$angle (vec{x}, vec{y}) = arccosbigg(frac{langle vec{x},vec{y}rangle}{sqrt{langle vec{x},vec{x}rangle} sqrt{langlevec{y},vec{y}rangle}}bigg)$$

The last definition makes sense because of the Cauchy-Schwartz inequality which is true for inner products: $$|langle vec{x},vec{y} rangle|^2 leq langlevec{x},vec{y} rangle langle vec{y}, vec{y}rangle$$

And hence, there exists a unique angle between any two vectors.

Now, by definition, an inner product is a bilinear form $langle cdot, cdot rangle: mathbb{R}^n times mathbb{R}^n to mathbb{R}$ such that

1. 
   $langle vec{x}, vec{x} rangle geq 0$ and $langle vec{x}, vec{x} rangle = 0$ if and only if $vec{x}=vec{0}$
   


2. $langle vec{x}, vec{y} rangle = langle vec{y}, vec{x} rangle$


3. $langle overrightarrow{alpha x+beta z}, vec{y} rangle = alphalangle vec{x}, vec{y} rangle + betalangle vec{z}, vec{y} rangle$

Again, by definition two vectors are perpendicular if and only if their inner product is zero. Accepting these definitions, the Pythagorean theorem can now be proven without any circular reasoning.

Now I understand that you feel kind of uncomfortable because you say that the standard inner product on $mathbb{R}^n$ comes from the Pythagorean theorem for $n=2,3$. And I understand that some people here interpret this as circular reasoning. However, I have two issues with this interpretation:

1. our definition for the standard inner product on $mathbb{R}^n$ for $n>3$ doesn't come from any previously known fact. Does this mean that our definition for $n>3$ is less valid than for $n=2,3$ or we should give up on higher dimensions?




2. It happens times and times in physics, as well as mathematics, that a new theory is based on some known facts in earlier theories as observations for a general pattern. Then to show that our new theory is better, they try to show that the results predicted by the previous theory can be obtained in the new framework as well. Should we say that it's circular reasoning? So, if someone applies a stronger theory to prove something that has motivated it in the earlier theory, is it circular reasoning? It seems that many people here think so. Even though I don't challenge it, I don't like this interpretation. I believe that once we have a new theory with its own definitions, which can be taught to a person without any prior knowledge of the earlier facts, it is not unreasonable to focus our attention only on our current framework and forget about its past like it never existed. Meanwhile, we can use our new theory to prove the results proven in our earlier theory.

If we start from Euclidean geometry in two dimensions, then it is circular. In this settings, we define perpendicular lines as lines that, at their intersection form four equal angles. Coordinate systems are defines using a basis consisting of two perpendicular vectors of the same length.

It is that actually easy to prove that two vectors are perpendicular iff their dot product is 0 using rotational properties. We would hit a snag, however, at the claim that the two-norm equals the length, which is a direct consequence of Pythagorean theorem.

Of course, that is not the context this proof of the Pythaogrean theorem normally shows up in. It shows up when we start from a different set of definitions: We define orthogonality using dot products. We define length using 2-norm. Then we prove a Vector space Pythagorean theorem as you have shown above, without circular reasoning.

Of course, now we have two different definitions for length and perpendicularity, and two different Pythagorean theorems. It is a separate step to prove that in $mathbb{R}^2$ and $mathbb{R}^3$, both sets of definitions actually mean the same things, and so both Pythagorean theorems are in fact the same theorem.