Proof using Permutation Symbols

[BlazNProdigy]

Homework Statement

Proove that...
(AxB)x(CxD)=(A.BxD)C-(A.BxC)D=(A.CxD)B-(B.CxD)A
using Permutation Symbols

Homework Equations

The Attempt at a Solution

I am confused about what to do after the third line from 'vela's response' (Post #2 from the reference link below).

Reference https://www.physicsforums.com/threa...rmutation-tensor-and-kroenecker-delta.454568/

 

[stevendaryl]

The problem is straight-forward, just tedious. You just apply the following rules:

1. $(X \times Y)^c = \epsilon_{abc} X^a Y^b$
2. $(X \cdot Y) = \delta_{ab} X^a Y^b$
3. $\epsilon_{abc} = \epsilon_{bca} = \epsilon_{cab} = -\epsilon_{bac} = -\epsilon{acb} = - \epsilon_{cba}$
4. $\epsilon_{abc} \epsilon_{ade} = \delta_{bd} \delta_{ce} - \delta_{be} \delta_{cd}$
5. $\epsilon_{abc} \delta_{ae} = \epsilon_{ebc}$

To get the ball rolling, rewrite what you're being asked to prove in terms of components:

$((A \times B) \times (C \times D))^c = (A \cdot (B \times D)) C^c - (A \cdot (B \times C)) D^c = (A \cdot (C \times D)) B^c - (B \cdot (C \times D)) A^c$