Solving a syestem of equations involving inner product

Question

Let $x=(x_1,x_2),y=(y_1,y_2), z=(z_1,z_2)\in \mathbb{R}^2$ be unit vectors with respect to an arbitrary inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{R}^2$. I am interested in the following system of equations for known $\alpha,\beta,\gamma\in \mathbb{R}$: $$\begin{cases}\langle x,y\rangle=\alpha\\ \langle x,z\rangle=\beta\\ \langle y,z\rangle=\gamma\end{cases}$$ I would appreciate any ideas.

Answer 1

Another point to be noted is that, you have three unit vectors in $\mathbb{R^2}$, so at least one is dependent on the other two. Say $z=ax+by$, then the last two equations can be combined to get

$$\langle x-y,z \rangle = \langle x-y,ax+by \rangle =a+\alpha b-\alpha a-b=\beta - \gamma.$$ Same as $$(\alpha -1)(b-a)=\beta - \gamma$$

This will be a condition for the system to be consistent.