Using Tensor Notations and Levi Civita Symbol to Prove Lagrange's Identity

[advphys]

Dear all,

Any idea for the proof of the Lagrange's identity using tensor notations and Levi Civita symbol?

(a x b).(c x d)=(a.c)(b.d) - (a.d)(b.c)

x: cross product
a,b,c,d: vectors

Thanks

 

[advphys]

Ok, thanks, in future i will be more careful.

What about the dot product on the left side, how can i use Levi Civita symbol to represent it.
Actually, the identity that you wrote and the cross product representation are all i know about the Levi Civita symbol but i couldn't use them.

 

[dx]

Use the following identity:

ε_ijkε_imn = δ_jmδ_kn - δ_jnδ_km

Also, in future, post questions like this in the homework section of PF, and tell us a little about how you've tried to solve the problem.

 

[dx]

Ok, thanks, in future i will be more careful.

What about the dot product on the left side, how can i use Levi Civita symbol to represent it.
Actually, the identity that you wrote and the cross product representation are all i know about the Levi Civita symbol but i couldn't use them.

The i component of a x b is a_jb_kε_ijk and the i component of c x d is c_md_nε_imn

So their dot product is a_jb_kc_md_nε_ijkε_imn

 

[advphys]

ok from there,
a_jc_jb_kd_k-a_jd_jb_kc_k
and i assume, similar form can be obtained for j and k components by just replacingg j s with k s, i s with j s and k s with i s. And in total i have 6 terms, 2 terms from each component. Am i right?

But, on the right had side i think i should have more than 6 terms?