Proof of Prop: Does a=b for all x in R3?

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Discussion Overview

The discussion centers around the proposition that if two vectors \( a \) and \( b \) in \( \mathbb{R}^3 \) satisfy the condition \( a \cdot x = b \cdot x \) for all vectors \( x \) in \( \mathbb{R}^3 \), then it follows that \( a = b \). Participants explore the validity of this proposition and its implications, particularly when considering the zero vector as a specific case.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the proposition holds true for all vectors \( x \) in \( \mathbb{R}^3 \) and specifically asks about the case when \( x \) is the zero vector.
  • Another participant emphasizes the importance of the phrase "for all vectors \( x \)" and suggests that examining the basis vectors could provide insight into the validity of the proposition.
  • A later reply presents an alternative proof, stating that \( a \cdot x = b \cdot x \) is equivalent to \( (a-b) \cdot x = 0 \), and argues that if this holds for all \( x \), then it leads to the conclusion that \( a = b \) by considering \( x = a-b \).

Areas of Agreement / Disagreement

Participants express differing views on the validity of the proposition, with some exploring specific cases and others providing proofs that support the proposition. The discussion remains unresolved, as no consensus is reached regarding the implications of the zero vector or the generality of the proposition.

Contextual Notes

Participants note the dependence on the definitions of vector operations and the implications of the condition being true for all vectors versus specific cases. There are also unresolved mathematical steps regarding the implications of the proofs presented.

nicolauslamsiu
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Prop: Suppose a and b be vectors in R3. If a·x=b·x for all vector x in R3, then a=b

My question if the proposition is always true.
And if x is a zero vector, is the proposition still valid?
 
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nicolauslamsiu said:
And if x is a zero vector, is the proposition still valid?
Note the words 'for all vectors x in R3', in the statement. Not 'for some vector x in R3'.

To see whether it's true for all pairs a,b, consider what you can conclude if it's true for all three of the basis vectors (1 0 0), (0 1 0), (0 0 1).
 
Aha, i have missed something. Thanks for ur help@@
 
Another proof: ##a \cdot x = b \cdot x## if and only if ##(a-b) \cdot x = 0##. If this holds for all ##x##, then choosing ##x = a-b## implies that ##(a-b)\cdot (a-b) = 0##. Since ##(a-b)\cdot (a-b)## is the square of the length of ##a-b##, this means that ##a-b## has length zero, so ##a-b = 0##, and therefore ##a=b##.
 

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