Binet–Cauchy identity
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In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin Louis Cauchy, states that
for every choice of real or complex numbers (or more generally, elements of a commutative ring). Setting ai = ci and bi = di, it gives the Lagrange's identity, which is a stronger version of the Cauchy-Schwarz inequality for the Euclidean space 
.
[edit] The Binet–Cauchy identity and exterior algebra
When n = 3 the first and second terms on the right hand side become the squared magnitudes of dot and cross products respectively; in n dimensions these become the magnitudes of the dot and wedge products. We may write it
where a, b, c, and d are vectors. It may also be written as a formula giving the dot product of two wedge products, as
[edit] Proof
Expanding the last term,
where the second and fourth terms are the same and artificially added to complete the sums as follows:
This completes the proof after factoring out the terms indexed by i. (q. e. d.)
