Is there an “inverted” dot product?

The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as:
$$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$

What about the quantity?
$$\mathbf{a} \star \mathbf{b} = \prod_{i=1}^{n} (a_{i} + b_{i}) = (a_{1} +b_{1})\,(a_{2}+b_{2})\cdots \,(a_{n}+b_{n})$$

Does it have a name?

“Dot sum” seems largely inappropriate. Come to think of it, I find it interesting that the dot product is named as such, given that it is, after all, a “sum of products” (although I am aware that properties of $\mathbf{a} \cdot{} \mathbf{b}$, in particular distributivity, make it a meaningful name).

$\mathbf{a} \star \mathbf{b}$ is commutative and has the following property:

$\mathbf{a} \star (\mathbf{b} + \mathbf{c}) = \mathbf{b} \star (\mathbf{a} + \mathbf{c}) = \mathbf{c} \star (\mathbf{a} + \mathbf{b})$

Answer

Too long for a comment, but I’ll list some properties below, in hopes some idea comes up.

  • ${\bf a}\star {\bf b}={\bf b}\star {\bf a}$;
  • $(c{\bf a})\star (c {\bf b})=c^n ({\bf a}\star {\bf b})$;
  • $({\bf a+b})\star {\bf c} = ({\bf a+c})\star {\bf b} = ({\bf b+c})\star {\bf a}$;
  • ${\bf a}\star {\bf a} = 2^n a_1\cdots a_n$;
  • ${\bf a}\star {\bf 0} = a_1\cdots a_n$;
  • $(c{\bf a})\star {\bf b} = c^n ({\bf a}\star ({\bf b}/c))$;
  • ${\bf a}\star (-{\bf a}) = 0$;
  • ${\bf 1}\star {\bf 0} = 1$, where ${\bf 1} = (1,\ldots,1)$;
  • $\sigma({\bf a}) \star \sigma({\bf b}) = {\bf a}\star {\bf b}$, where $\sigma \in S_n$ acts as $\sigma(a_1,\ldots,a_n) \doteq (a_{\sigma(1)},\ldots,a_{\sigma(n)})$.

Attribution
Source : Link , Question Author : doc , Answer Author : Ivo Terek

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