For two inner products, can we find or proof a relation between them by a mathematical property?

Question

I want to know if there is any method or property that can ensure a relation between two inner products; For example we have three vectors x, y, and z and we need to ensure the following relation :
$\langle x,y \rangle. \langle z,y \rangle = \langle x,z \rangle$

Answer 1

If you want the condition to hold for all vectors $x,y,z$, then you're out of luck.

Specializing to $x=y=z$, the equation says $\langle x,x\rangle^2 = \langle x,x \rangle$. Since only the $0$ vector is allowed to have $\langle 0, 0 \rangle = 0$, this means that $\langle x,x \rangle = 1$ for every nonzero vector $x$.

I will leave it to you to show that this condition is not satisfiable (hint: consider $\langle \lambda x, \lambda x \rangle$ for some nonzero scalar $\lambda$).

---

I hope this helps ^_^

Answer 2

The equation holds iff $x$ is orthogonal to $ \langle z, y \rangle y-z$.