- #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?