Proving the Theorem: A, B, C, and D Vectors | Step-by-Step Guide

[rado5]

Homework Statement

How can I prove this theorem? A, B, C and D are vectors.

(A$$\times$$B).(C$$\times$$D)=(A.C)(B.D)-(A.D)(B.C)

Homework Equations

A$$\times$$B=ABsin($$\theta$$) and A.B=ABcos($$\theta$$)

The Attempt at a Solution

Please help me solve it.

 

[aPhilosopher]

There's one more relevant equation. As a hint, think of AxB as an additional vector V.

 

[VeeEight]

This can get messy, but have you considered the coordinate notation, that is for vectors A = <a₁, a₂, a₃>, B = <b₁, b₂, b₃>, A x B = <a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁>
(http://en.wikipedia.org/wiki/Cross_product#Matrix_notation)

 

[rado5]

This can get messy, but have you considered the coordinate notation, that is for vectors A = <a₁, a₂, a₃>, B = <b₁, b₂, b₃>, A x B = <a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁>
(http://en.wikipedia.org/wiki/Cross_product#Matrix_notation)

yeah I'm sure that this solves the problem, and it seems obvious, so thank u very much, but do u know an easier way or a cleaner (lol) way to prove it.