- #1

dingo_d

- 211

- 0

## Homework Statement

Basically I have reciprocal lattice vectors:

[tex]a'=\frac{b\times c}{a\cdot(b\times c)}[/tex]

[tex]b'=\frac{c\times a}{a\cdot(b\times c)}[/tex]

[tex]c'=\frac{a\times b}{a\cdot(b\times c)}[/tex]

And I have to prove that these relations hold:

[tex]a=\frac{b'\times c'}{a'\cdot(b'\times c')}[/tex]

[tex]b=\frac{c'\times a'}{a'\cdot(b'\times c')}[/tex]

[tex]c=\frac{a'\times b'}{a'\cdot(b'\times c')}[/tex]

## The Attempt at a Solution

I really dk where to start :\

Do I try with the direct component expansion or can I do it with Levi-Civita symobol:

[tex]a\cdot(b\times c)=\varepsilon_{ijk}a_ib_jc_k[/tex] and [tex]b\times c=\varepsilon_{ijk}b_jc_ke_i[/tex]

And then it would be:

[tex]a'=\frac{\varepsilon_{ijk}b_jc_ke_i}{\varepsilon_{ijk}a_ib_jc_k}[/tex]

but what can I do with it?