Proofs in analytic geometry and vector spaces.

  • #1
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Homework Statement:

All below

Relevant Equations:

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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
1594786685858.png

1594786698232.png
(the rest is easy)

What seems to me more general proof.
 

Answers and Replies

  • #2
Delta2
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There is indeed a problem with #2, you have to prove that it holds in any base but i dont understand the problem with #1, how are you supposed to work with vectors if you dont 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.
 

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