Proof involving cross products

[lemmiwinks]

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.

 

[rock.freak667]

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.

 

[lemmiwinks]

Yeah I guess it was a stupid question I was just trying to think of how I would write it down on paper.

 

[rock.freak667]

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.

 

[Edgardo]

Since you mentioned tensors and indices, are you supposed to use the http://folk.uio.no/patricg/teaching/a112/levi-civita/index.html" ?

 

[LCKurtz]

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.