# Proofs in analytic geometry and vector spaces.

• LCSphysicist
In summary, the conversation discusses different methods of proving that the diagonals of a square are orthogonal. One method involves using vectors and their coordinates, while the other involves using plane geometry. The latter method is considered to be more general and complete, but there is a potential issue with one of the assumptions made in the proof. The conversation also touches on the importance of adapting a vector space in order to work with vectors.
LCSphysicist
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I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about:

For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?:

d1 = Diagonal one = (a,b,c)
d2 = Diagonal two =(-a,-b,c)

but a² + b² = l² = c²

d1*d2 = (-a² -b² + c²) = (-(a² + b²) + c²) = > This is the dot product

The problems i see is:

#1 = I adopt a R3 space.
#2 = I assumed the basis orthogonal, so the distance between the vertex lying in the same side is (a²+b²)^(1/2)

The dot product holds for any basis anyway.

Can someone say: "(This is not a general proof, you adopt the orthogonal basis and R3)"?

We could try to proof by plane geometry, this is the general proof?

OBS: There is too

(the rest is easy)

What seems to me more general proof.

There is indeed a problem with #2, you have to prove that it holds in any base but i don't understand the problem with #1, how are you supposed to work with vectors if you don't adapt a vector space(R^3 or R^2)?

You are right that the proof with u and v is more general more complete i would say.

## 1. What is analytic geometry?

Analytic geometry is a branch of mathematics that combines algebra and geometry to study geometric shapes and their properties using coordinate systems. It involves the use of equations and coordinates to represent geometric figures and solve problems related to them.

## 2. What are proofs in analytic geometry?

Proofs in analytic geometry are used to logically demonstrate the validity of a statement or theorem related to geometric figures. They involve using mathematical principles and reasoning to show that a given statement is true.

## 3. How are proofs in analytic geometry different from other types of proofs?

Proofs in analytic geometry are different from other types of proofs because they involve the use of coordinates and equations to represent geometric figures, rather than just using geometric constructions. This allows for a more algebraic approach to solving geometric problems.

## 4. What are vector spaces in analytic geometry?

Vector spaces in analytic geometry are mathematical structures that consist of a set of vectors and operations (such as addition and scalar multiplication) that can be performed on those vectors. They are used to represent and study geometric concepts such as lines, planes, and higher dimensional spaces.

## 5. How are vector spaces used in proofs in analytic geometry?

Vector spaces are used in proofs in analytic geometry to show the properties and relationships between geometric figures. They allow for a more abstract and general approach to solving geometric problems, making it easier to prove theorems and statements about geometric shapes and their properties.

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