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

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Homework Help Overview

The discussion revolves around the vector identity Ax(BxC) and its validity under certain conditions regarding the vectors A, B, and C. Participants explore whether the identity holds when the vectors are equal or not.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants question the conditions under which the identity holds, particularly focusing on the implications of having A equal to B or C. There is also a discussion about the nature of identities in mathematics and their universality.

Discussion Status

The conversation is ongoing, with some participants providing insights into the nature of vector identities while others express concerns about the clarity of assumptions in various sources. There is no explicit consensus reached on the validity of the identity under specific conditions.

Contextual Notes

Participants note that some sources may not clearly state the conditions under which the identity is valid, leading to confusion about its applicability when vectors are equal.

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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