Vector Proof

1. Nov 10, 2013

thercias

1. The problem statement, all variables and given/known data
if A' = BxC/(A.BxC)
B' = CxA/(A.BxC)
C' = AxB/(A.BxC)

prove that A = B'xC'/(A'.B'xC')
B = C'xA'/(A'.B'xC')
C=(A'xB')/(A'.B'xC')

2. Relevant equations

3. The attempt at a solution
I was sort of lost on how to do this. First I tried to simplify a', b', and c' and plug those into a, b, c but that didn't work out.
then i took the dot product of A' with A
so A'.A = A.(BxC)/(A.BxC) = 1
doing the same for B' and C' makes the answer 1 too. then i did the dot product of A with A' and got 1, and same result for B and C. I'm not sure if that means anything, but now I'm kind of stuck on how to solve this.

2. Nov 11, 2013

tiny-tim

hi thercias!

do you know the formula for A x (B x C) ?

(if not, look it up!)

apply that to the numerator (top) of B' x C' …

show us what you get

3. Nov 11, 2013

hilbert2

The vectors in that problem are defined exactly the same way as the reciprocal lattice vectors in crystallography and solid state physics. In the problem you're basically being asked to show that the reciprocal of the reciprocal lattice is the original lattice...

4. Nov 11, 2013

thercias

Hi, yes I do know the formula for AxBxC. So if we use that for a I get
(A'x(B'xC'))/(A'.B'xC')
=(B'(A'.C')-C'(A'.B'))/(A'.B'xC')
=B'(A'.C')/(A'.B'xC') - C'(A'.B')/(A'.B'xC')

but this looks kind of messy doesn't it? I'm not quite sure how this will lead to a solution.

5. Nov 11, 2013

tiny-tim

hi thercia!
what are you doing??

you need to convert B'xC' into A B and C ​

6. Nov 11, 2013

thercias

Truthfully, I'm not really sure. So If I'm following correctly, I convert B'xC' into
CxA/(A.BxC) x AxB/(A.BxC)
but now I'm stuck on how you're supposed to apply Ax(BxC) on this part. I'm not sure if I'm doing something wrong again.
am I supposed to find CxA x AxB?

Last edited: Nov 11, 2013
7. Nov 11, 2013

tiny-tim

(CxA) x (AxB)

[put CxA = P]

= P x (Q x R) = (P.R)Q etc

8. Nov 11, 2013

thercias

P x (A x B)
= A(P.B) - B(P.A)

so b'xc' = (A(CxA.B)-B(CxA.A))/(A.BxC)
= A(CxA.B)/(A.BxC) - B(CxA.A)/(A.BxC)
CxA.B and A.BxC are the same so they cancel
= A-B(CxA.A)/(A.BxC) I'm not sure what relationship CxA.A and A.BxC have but I probably have to simplify that somehow.

9. Nov 11, 2013

tiny-tim

think!

CxA.A = … ?

10. Nov 11, 2013

thercias

hm, isn't it just zero? now that i think about it it can be represented by a determinant, and it will have repeated rows making it 0.

so b'xc' = A

so a = A/(A'.B'xC')
=A/(A'.A)

and a' = A'/(A.A')

11. Nov 11, 2013

tiny-tim

that's it, isn't it?

12. Nov 11, 2013

thercias

I guess it is! Thanks for the help.
I noticed that if you sub a' into
a= a/(a.a') we get
a=a(a.bxc)/(a.bxc)
a=a