What is the validity of the vector identity Ax(BxC)?

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SUMMARY

The vector identity Ax(BxC) is valid for any choice of vectors A, B, and C, including cases where A equals B or C. The identity is expressed as Ax(BxC) = B(A dot C) - C(A dot B) and holds true regardless of the equality of the vectors involved. It is crucial to understand that while some mathematical identities have specific conditions, this particular identity is universally applicable across all vector selections.

PREREQUISITES
  • Understanding of vector operations, specifically cross products and dot products.
  • Familiarity with vector identities and their derivations.
  • Basic knowledge of mathematical notation and terminology related to vectors.
  • Awareness of conditions under which certain mathematical identities hold true.
NEXT STEPS
  • Study the derivation of the vector identity Ax(BxC) in detail.
  • Explore other vector identities and their conditions for validity.
  • Learn about the implications of vector equality in cross product operations.
  • Investigate the differences between universal identities and those with specific conditions.
USEFUL FOR

Students of mathematics, physics enthusiasts, and anyone studying vector calculus or linear algebra will benefit from this discussion.

omegacore
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Homework Statement



Regarding the identity Ax(BxC)

Homework Equations



Does this identity only hold when A != B != C?
 
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Which identity are you referring to? Whatever the identity it would work for any vector...but of course the cross product of 2 equal vectors is zero =)
 
Ah yes, I forgot the identifying portion of the identity:

Ax(BxC) = B(A dot C) - C(A dot B)

Same qualifying question as before. Obviously this identity does not just fall out of the sky and is the product of a process. I am wondering if the process is disrupted (invalid identity) by having A = B... it seems like it wouldn't be.
 
No it wouldn't, I wonder what makes you think so?
 
It's fair to wonder, because some sources tend to be somewhat sloppy about explicitly stating hypotheses.
 
An identity holds for any choice of vectors. That's what makes it an "identity".
 
Not all identities are universal. For example,
sin arcsin x = x​
is only valid on the interval [-\pi/2, \pi/2].
 

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