Transitive Property with Orthogonal Vectors?

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SUMMARY

The discussion centers on the question of whether the transitive property holds for orthogonal vectors in R^3. Specifically, it examines if x1 being orthogonal to x2 and x2 being orthogonal to x3 implies that x1 is orthogonal to x3. The consensus is that this is not necessarily true, as demonstrated through geometric reasoning and the need for algebraic proof. The participant expresses uncertainty in proving the relationship algebraically, indicating a gap in understanding the properties of orthogonal vectors.

PREREQUISITES
  • Understanding of vector spaces, specifically R^3.
  • Knowledge of orthogonality in linear algebra.
  • Familiarity with matrix transposition and dot product operations.
  • Basic geometric interpretation of vectors and their relationships.
NEXT STEPS
  • Study the properties of orthogonal vectors in linear algebra.
  • Learn about the geometric interpretation of vector relationships.
  • Explore algebraic proofs involving dot products and vector orthogonality.
  • Investigate counterexamples that illustrate non-transitive properties in vector spaces.
USEFUL FOR

Students of linear algebra, educators teaching vector properties, and anyone interested in the geometric and algebraic aspects of orthogonality in vector spaces.

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Homework Statement



Let x1, x2, and x3 be vectors in R^3. If x1 is orthogonal to x2 and x2 is orthogonal to x3, is it necessarily true that x1 is orthogonal to x3?


Homework Equations



I know that if x1 is orthogonal to x2 and x2 is orthogonal to x3, then...

(x1)^T*x2=0
(x2)^T*x3=0


The Attempt at a Solution



I think that the answer would be no. I can imagine it geometrically, but I'm not sure how I would prove this algebraically.

I would obviously have to prove that (x1)^T*x3 would ALWAYS have to equal 0 if it the statement was necessarily true... but I'm not sure how to go about doing that...

I'm completely stuck. I would greatly appreciate your help!
 
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