Let's assume $v,w, x_i \in R^n$ are unknown. Can one compute dot product $\langle v,w\rangle$ if one has just the numbers: $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$ for $n$ random vectors $x_i$.

If $x_i = e_i$ it is quite simple: $$ \langle v,w\rangle = \sum_i \langle v,e_i\rangle \langle w,e_i\rangle$$

But what if the $x_i$ is not an orthogonal basis ?

If not possible can we do something with stronger assumptions like all vectors being unitary ?