Vector relationship? |A+B| = |A-B|

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SUMMARY

The discussion centers on the vector relationship defined by the equation |A+B| = |A-B|, indicating that vectors A and B have equal magnitudes regardless of their directions. Participants explore the implications of this relationship using the inner product, specifically calculating norms with the formula |A|^2 = . The conversation highlights the derivation of the equation through the properties of the dot product, leading to the conclusion that must equal zero, indicating that vectors A and B are orthogonal. The notation for the dot product is also a point of confusion among participants.

PREREQUISITES
  • Understanding of vector norms and magnitudes
  • Familiarity with inner product notation
  • Knowledge of properties of the dot product
  • Basic concepts of orthogonality in vector spaces
NEXT STEPS
  • Study the properties of vector norms and their implications
  • Learn about orthogonal vectors and their significance in linear algebra
  • Explore inner product spaces and their applications
  • Review examples of vector relationships in physics and engineering contexts
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Students and professionals in mathematics, physics, and engineering who are working with vector analysis and seeking to deepen their understanding of vector relationships and inner product spaces.

Fjolvar
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I've been spending far too much time on this problem and I know I'm over thinking it. Here it is:

If |A+B| = |A-B|

What is the most general relationship between the two vectors?

-Now I know this is just saying they have equal magnitude regardless of direction, but I'm not quite sure what it's asking for. What kind of general relationship am I supposed to write out? Any help would be greatly appreciated. Thanks!
 
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Try to calculate this norms with the inner product:

|A|^2=<A,A>
 
micromass said:
Try to calculate this norms with the inner product:

|A|^2=<A,A>

I'm still not seeing how to relate vector A and B using this.. =/
 
What did you get when you wrote it out:

|A+B|^2=<A+B,A+B>=...

|A-B|^2=...

?
 
<A+B,A+B>=<A-B,A-B>
By properties of the dot product..
<A,A>+2<B,A>+<B,B>=<A,A>-2<B,A>+<B,B>

Get everything to one side and deduce from that.
 
We haven't learned this notation yet unfortunately. Is this the only possible approach?
 
Fjolvar said:
We haven't learned this notation yet unfortunately. Is this the only possible approach?

What is yor notation for the dot product then??
 

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