- #1
physicsss
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Show that x+y and x-y are orthogonal if and only if x and y have the same norms.
Can someone get me started?
Can someone get me started?
Show x+y & x-y are Orthogonal if x & y Have Same Norms
- Context: Undergrad
- Thread starter physicsss
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- Orthogonal Vectors
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Discussion Overview
The discussion centers on the relationship between the vectors x+y and x-y, specifically exploring the conditions under which they are orthogonal, and how this relates to the norms of the vectors x and y. The scope includes mathematical reasoning and geometric interpretations.
Discussion Character
- Mathematical reasoning
- Geometric application
- Exploratory
Main Points Raised
- One participant seeks clarification on the definition of "orthogonal" in this context.
- Another participant suggests that if x+y and x-y are orthogonal, then it must imply a relationship between the norms of x and y.
- A participant proposes a geometric interpretation, questioning if the orthogonality condition leads to an isosceles right triangle formed by the vectors x, y, and x+y.
- It is noted that the equation (x+y)(x-y)=0 can be transformed into ||x||^2 = ||y||^2, leading to the conclusion that ||x|| = ||y||.
- A later reply states that the condition of having perpendicular diagonals in a parallelogram indicates that the vectors have equal magnitudes, suggesting a reverse proof of a geometric theorem through algebraic means.
Areas of Agreement / Disagreement
Participants express various interpretations and applications of the orthogonality condition, but there is no consensus on a singular interpretation or conclusion regarding the geometric implications.
Contextual Notes
Some assumptions about the definitions of orthogonality and norms are not explicitly stated, and the discussion does not resolve the implications of the geometric interpretations presented.
Who May Find This Useful
Readers interested in vector mathematics, geometric interpretations of algebraic results, and the properties of orthogonal vectors may find this discussion relevant.
- #2
HallsofIvy
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1) What does "orthogonal" mean here?
2) So if x+y and x-y are orthogonal what must be true?
3) And in order for that to be true what about x and y?
2) So if x+y and x-y are orthogonal what must be true?
3) And in order for that to be true what about x and y?
- #3
dextercioby
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Can you think of a nice geometrical application of this result
[tex]\left\langle x+y, x-y\right\rangle =0 \Longleftrightarrow ||x||=||y||[/tex] ?
Daniel.
[tex]\left\langle x+y, x-y\right\rangle =0 \Longleftrightarrow ||x||=||y||[/tex] ?
Daniel.
- #4
physicsss
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So (x+y)(x-y)=0, which can be turned into ||x||^2 = ||y||^2 take the square root of each side, I get ||x|| = ||y||.
As for dextercioby's question, if that is true, then x, y, and x+y make up an isosceles right triangle?
As for dextercioby's question, if that is true, then x, y, and x+y make up an isosceles right triangle?
- #5
dextercioby
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The paralelelogramme with perpendicular (onto another) diagonals is a rhombus. Therefore, the vectors have equal modulus. Actually, u've proven the reverse, viz.the geometrical result (theorem/proposition) by algebraic methods only. 
Daniel.
Daniel.
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