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Given vectors $x,y,z \in \mathbb{R}^n$, define that their ternary dot product $x \cdot y \cdot z$ is the following real number:

$$\sum_{i=1}^n x_i y_i z_i$$

Recall that the usual (binary) dot product has a lot of geometric meaning. For example the dot product of two vectors is $0$ iff they’re orthogonal. My question is: does the ternary dot product have any geometric significance? For example, suppose $x \cdot y \cdot z$ equals $0$. Does this condition have a geometric interpretation?

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If it has a geometric interpretation, that interpretation certainly has to be less obvious. One major setback seems to be, that the *ternary dot product* is NOT invariant under rotations. Consider the example $x,y,z\in\mathbb R^2$ given by

$$

x=\begin{pmatrix}1\\0\end{pmatrix},\quad y=\begin{pmatrix}\sqrt 2/2\\\sqrt 2/2\end{pmatrix},\quad\text{and }z=\begin{pmatrix}0\\1\end{pmatrix}

$$

which has ternary dot product equal to zero and may be rotated by $\pi/4$ to

$$

x’=\begin{pmatrix}\sqrt 2/2\\\sqrt 2/2\end{pmatrix},\quad y’=\begin{pmatrix}0\\1\end{pmatrix},\quad\text{and }z’=\begin{pmatrix}-\sqrt 2/2\\\sqrt 2/2\end{pmatrix}

$$

having ternary dot product $\langle x’,y’,z’\rangle=1/2$.

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