Proof involving cross products

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Homework Help Overview

The discussion revolves around proving a vector identity involving cross products and dot products within the context of tensors and indicial notation. The original poster presents a curiosity about the relationship between three perpendicular vectors, A, B, and C, and seeks to establish that the dot product of the cross products (A × C) · (B × C) equals zero.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the properties of cross products and the implications of perpendicular vectors. Questions arise about the nature of the vectors resulting from the cross products and their relationships to the original vectors.

Discussion Status

The discussion includes various attempts to articulate the proof and reasoning behind the identity. Some participants suggest exploring the geometric interpretations of the vectors involved, while others offer insights into the relationships between the vectors without reaching a consensus on a formal proof.

Contextual Notes

Participants note the context of the problem as being related to engineering studies and tensor mathematics, indicating a level of complexity and specificity in the discussion. There is also mention of potential resources or methods, such as the Levi-Civita symbol, which may be relevant to the proof.

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So I'm an engineering student and we're doing some work with tensors and indicial notation, and I came across something that I know is true but couldn't think of how to prove. I don't need it for homework or anything it's just a curiosity thing. OK, so

Basically take a set of axes, 3 perpendicular vectors, call them A B and C
Prove (AXC)'dot'(BXC) = 0 (ie the vectors are perpendicular, X stands for cross product)

It seems like it should be really obvious but I can't think of how to solve it like a proof... I'm probably going to feel like a moron when somebody answers but whatever.
 
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If A,B and C are perpendicular.

What vector does AxC give? What vector does BxC give? Knowing that when when you find the cross-product of two vectors, you get a vector perpendicular to the plane containing the two crossed vectors.
 
Yeah I guess it was a stupid question I was just trying to think of how I would write it down on paper.
 
lemmiwinks said:
Yeah I guess it was a stupid question I was just trying to think of how I would write it down on paper.

Well you could just write it as AxC =B and BxC=A, B.A = 0. You can probably expand (AxC).(BxC) and get it out. But that takes too much time in case you don't know what a.(bxc) equals.
 
Since you mentioned tensors and indices, are you supposed to use the http://folk.uio.no/patricg/teaching/a112/levi-civita/index.html" ?
 
Last edited by a moderator:
lemmiwinks said:
So I'm an engineering student and we're doing some work with tensors and indicial notation, and I came across something that I know is true but couldn't think of how to prove. I don't need it for homework or anything it's just a curiosity thing. OK, so

Basically take a set of axes, 3 perpendicular vectors, call them A B and C
Prove (AXC)'dot'(BXC) = 0 (ie the vectors are perpendicular, X stands for cross product)

It seems like it should be really obvious but I can't think of how to solve it like a proof... I'm probably going to feel like a moron when somebody answers but whatever.

A X C is perpendicular to the plane of A and C so is parallel to B. B X C is perpendicular to B so it is perpendicular to A X C. That's why the dot product is 0.
 

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