Binet–Cauchy identity

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 "a_i" = "c_i" and "b_i" = "d_i", it gives the Lagrange's identity, which is a stronger version of the Cauchy-Schwarz inequality for the Euclidean space $scriptstylemathbb\{R\}^n$.

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

:$(a\; cdot\; c)(b\; cdot\; d)\; =\; (a\; cdot\; d)(b\; cdot\; c)\; +\; (a\; wedge\; b)\; cdot\; (c\; wedge\; d),$

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

:$(a\; wedge\; b)\; cdot\; (c\; wedge\; d)\; =\; (a\; cdot\; c)(b\; cdot\; d)\; -\; (a\; cdot\; d)(b\; cdot\; c).,$

Proof

Expanding the last term,

::

where the second and fourth terms are the same and artificially added to complete the sums as follows:

:$=sum\_\{i=1\}^n\; sum\_\{j=1\}^na\_i\; c\_i\; b\_j\; d\_j-sum\_\{i=1\}^n\; sum\_\{j=1\}^na\_i\; d\_i\; b\_j\; c\_j.$

This completes the proof after factoring out the terms indexed by "i". "(q. e. d.)"