Simple dot product [inner product] (verification)

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SUMMARY

The discussion centers on verifying the properties of the dot product, specifically in the context of a homework problem involving vectors u and w. The key insight provided is that for part A, the expression (u-w)·v = 0 for all vectors v indicates that u must equal w. This conclusion is essential for understanding the implications of the dot product in linear algebra. The clarification that a specific example suffices for part B, but not for part A, highlights the difference in requirements for proving vector equality versus demonstrating a property.

PREREQUISITES
  • Understanding of vector operations, specifically the dot product.
  • Familiarity with linear algebra concepts, particularly vector equality.
  • Knowledge of mathematical proofs and verification techniques.
  • Ability to interpret and manipulate vector equations.
NEXT STEPS
  • Study the properties of the dot product in linear algebra.
  • Learn about vector spaces and their dimensions.
  • Explore mathematical proof techniques, focusing on vector equality.
  • Investigate applications of the dot product in physics and engineering.
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Students studying linear algebra, educators teaching vector mathematics, and anyone interested in the theoretical foundations of vector operations.

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Homework Statement


2v1m3ax.png



Homework Equations



know dot product

The Attempt at a Solution


PART A
2cr2ij6.png


PART B
not sure what's it asking for

help would be great
 
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Well, what you posted is the answer to the B part actually :wink: Notice the "for all v" part in A. Specific example of u,v does the trick for B, not for A. Hint for A: (u-w)*v = 0 for all v. What does it say about u - w?
 

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