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 $\alpha ,\beta \in \mathbb{F}\setminus \{0\}$. We study the quantity $$\mathrm{\mid }{\mathrm{\Pi }}_{\alpha ,\beta }\mathrm{\mid }=\{(p,q,r)\in P\times P\times P\mid p\cdot q=\alpha ,p\cdot r=\beta \}\mathrm{.}$$ We observe a connection between the question of placing an upper bound on $\mathrm{\mid }{\mathrm{\Pi }}_{\alpha ,\beta }\mathrm{\mid }$ 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 $\mathrm{\mid }{\mathrm{\Pi }}_{\alpha ,\beta }\mathrm{\mid }$ in a variety of settings.