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I know that the dot product of two vectors is the sum of element-wise multiplication. Using pseudo-MATLAB notation:

(x,y) = sum(x.*y).

I'm interested in computing something analogous for three vectors:

(x,y,z) = sum(x.*y.*z).

Is there a mathematical definition or formalism for this ternary operation?

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  • $\begingroup$ I don't think this operation has a name, nor that it has interesting properties. $\endgroup$ – Crostul Aug 19 '15 at 19:09
  • $\begingroup$ You can certainly compute this, but the dot product has an actual meaning-it's used to compute the angle between two vectors. $\endgroup$ – Kevin Carlson Aug 19 '15 at 19:09
  • $\begingroup$ Even without knowing your actual context, probably you want $x\cdot (y \times z)$, that is the dot product of $x$ with the "cross product" of $y$ and $z$. This is a reasonable and useful thing. Also, as it stands, there's sort of a "type error" insofar as I don't know what that alleged ternary operation could mean, since the vectors/matrices are not the right size to be multiplied literally as the notation suggests. $\endgroup$ – paul garrett Aug 19 '15 at 20:11
  • $\begingroup$ @wonderboy3489 You just defined and formalized the idea. Whether there is any literature on the subject is a different question. A brief search of "ternary dot product" turns up math.stackexchange.com/questions/1196736/… $\endgroup$ – DanielV Aug 19 '15 at 20:20
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You could call it, for example, ${\bf x}^T Y {\bf z}$ where $Y$ is the diagonal matrix with the entries of ${\bf y}$ on the diagonal. Of course you could apply any permutation to ${\bf x}$, ${\bf y}$, ${\bf z}$.

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Your "ternary dot product" is basically equivalent to the Hadamard product (https://en.wikipedia.org/wiki/Hadamard_product_%28matrices%29): given two vectors $\langle a_i\rangle, \langle b_i\rangle$, their Hadamard product is just componentwise multiplication: $$\langle a_i\rangle*\langle b_i\rangle=\langle a_ib_i\rangle.$$ To see why this is equivalent to the thing you describe, note that the ternary dot product of $\alpha, \beta, \gamma$ is just $\alpha\cdot (\beta*\gamma)$. In fact, the Hadamard product can be used to define $n$-ary dot products similarly, whereas you can't obviously do this with just the ternary dot product alone. Because of this, I don't think this operation deserves a specific name.

Now, the Hadamard product probably seems really natural (it certainly did to me), so you might ask, "Why isn't the Hadamard product more ubiquitous?" Basically, see why don't we define vector multiplication component-wise?, especially the answer by Giuseppe Negro; essentially, equations in terms of the Hadamard product aren't invariant under rotations, so it is unlikely to have any phsyical meaning (if something has physical meaning then computations involving it should be independent of exactly how we draw our axes). By contrast, equations in terms of the usual dot product are invariant under a wide range of transformations.

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  • $\begingroup$ In fact the only isomorphisms that preserve this triple-product are the permutations of the basis vectors. To see this note that the basis vectors are the only vectors with $(v,-,-)$ of rank $1$ and $(v,v,v)=1$. (At least this is true over $\mathbb R$. Over $\mathbb C$ there is also the map $\omega I$ where $\omega^3=1$.) $\endgroup$ – Oscar Cunningham Aug 24 '15 at 17:38

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