What are the Conditions for Equality in Cauchy and Triangle Inequalities?

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Discussion Overview

The discussion revolves around the conditions under which equality holds in the Cauchy and Triangle inequalities. Participants explore the implications of different types of numbers (real, complex, vectors) on these inequalities.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants inquire about the specific conditions for variables a and b that would allow equality in both the Cauchy and Triangle inequalities.
  • One participant suggests that the inequalities can be expressed as functions of a and b, indicating a need for clarity in the definitions used.
  • Another participant states that Cauchy's inequality can be viewed as the trivial case of |a . b| <= |a||b| and reiterates the Triangle inequality as |a+b| = |a| + |b|.
  • A question is raised regarding the behavior of the inequalities when both a and b are positive or both are negative, suggesting that the sign of a and b may influence the conditions for equality.
  • One participant advises reviewing the proof of the inequalities for insights into the conditions for equality.
  • There is a reiteration of the need to specify the nature of a and b (real numbers, complex numbers, vectors) to properly address the question of equality.

Areas of Agreement / Disagreement

Participants express varying viewpoints on the conditions for equality in the inequalities, and no consensus is reached regarding the specific requirements or implications based on the types of numbers involved.

Contextual Notes

The discussion lacks clarity on the definitions of a and b, and the implications of their types (real, complex, vectors) remain unresolved. The proofs of the inequalities are referenced but not detailed, leaving some mathematical steps unaddressed.

Design
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I was wonder what conditions a and b have to be for each inequality in order to satifsy the equality?
 
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Design said:
I was wonder what conditions a and b have to be for each inequality in order to satifsy the equality?

It would help if you wrote out the inequality as a function of a and b. I'm not a mind reader.
 
Cauchy's inequality is just the trivial |a . b| <= |a||b| Oo

Triangle inequality

|a+b| = |a| +|b|
 
What do you think happens when a,b are both positive or both negative ?
 
Just go over the proof, and you'll probably see it.
 
Design said:
Cauchy's inequality is just the trivial |a . b| <= |a||b| Oo

Triangle inequality

|a+b| = |a| +|b|

To answer your question, you need to specify what a and b are. Possibilities: real numbers, complex numbers, vectors (either real or complex).
 

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