Geometric Products
Generalized geometric product
Recalling that for k ≥ 1, the outer product of a vector v with a k-vector K is a (k + 1)-vector defined in terms of the geometric product by
v ∧ K = 1⁄2 (vK + (−1)k Kv) = (−1)k K ∧ v (8)
and the corresponding inner product
v ・ K = 1⁄2 (vK + (−1)k+1 Kv) = (−1)k+1 K ・ v (9)
Adding (8) and (9) we obtain
vK = v ・ K + v ∧ K (10)
which obviously generalizes (7). The important thing about (10), is that it decomposes vK into (k −1) - vector and (k +1) - vector parts.