I'll begin with the specific case I am interested and then ask a more general question.

Suppose $x_1,\ldots,x_4$ are a collection of (unknown) unit vectors in $\mathbb{R}^2$. Given $\left<x_1,x_2\right>,\left<x_2,x_3\right>,\left<x_3,x_4\right>,\left<x_4,x_1\right>$ is the value of $\left<x_1,x_3\right>$ uniquely determined? I'm essentially positive they are with some algebra, linear algebra, and a proof by image, but I'm looking for a more elegant way to solve this and/or results that yield a more straightforward way to show this.

The above can be considered like a square graph, with edges between two vectors indicating a known inner product. Given the same situation, with different dimension and/or different number of vectors, can we determine the uniqueness of all inner products given the graph structure?

Edit: I'm wondering if this can be nicely formalized as a graph rigidity problem... I'm also interested in the case where the norm of each $x_i$ is unknown, other than being nonzero.