# dot product of vectors with noncommutative elements

I came upon an interesting problem. Suppose you have three vectors $$\psi=(a, b)$$, $$\phi=(c,d)$$ such that their components do not commute: $$ac\neq ca$$, $$bd \neq db$$ etc. (in general the vectors may have more than two components) How do you perform the inner product ψ.φ?

• You could use Inner[] with NonCommutativeMultiply[]... – J. M.'s ennui Dec 23 '20 at 11:03
• @J.M.'sdiscontentment Inner allows the calculation of the inner product of only two vectors. If I need the inner product of 4 or 6? – geom Dec 23 '20 at 12:52
• The inner product is a scalar and can be factored out. So for e.g. 4 vectors it would read: v1.v2 v3.v4 – Daniel Huber Dec 23 '20 at 13:00
• By "." I mean any inner product you are using. – Daniel Huber Dec 23 '20 at 13:36
• How exactly do you define the inner product of three or more vectors? – J. M.'s ennui Dec 23 '20 at 14:19