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Incidences and pairs of dot products

Let $\mathbb{F}$ be a field, let $P \subseteq \mathbb{F}^d$ be a finite set of points, and let $α,β\in \mathbb{F} \setminus \{0\}$. We study the quantity \[|Π_{α, β}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q = α, p \cdot r = β\}.\] We observe a connection between the question of placing an upper bound on $|Π_{α,β}|$ and a well-studied question on the number of incidences betwen points and hyperplanes, and use this connection to prove new and strengthened upper bounds on $|Π_{α,β}|$ in a variety of settings.