What can we infer from (x|y)=(x|z) in a real inner product space?

Question

$(x|y) = (x|z)$

I think the vectors $y$ and $z$ are parallel to each other.

What else can be said?

Answer 1

I think you are talking about $$ \langle x,y \rangle=\langle x,z \rangle $$ being true for any $x,y,z \in (V, \langle\cdot,\cdot\rangle)$, then you can not infer $y$ and $z$ being parallel to each other. A counterexample would be $V=\mathbb{R}^3$ and $x=(1,0,0)^t, y=(0,1,0)^t$ and $z=(0,0,1)^t$. Even though with the standard inner product $$ \langle x,y \rangle=1\cdot0+0\cdot1+0\cdot0=0=1\cdot0+0\cdot0+0\cdot1=\langle x,z \rangle $$ your identity holds, it is obviously not true that $y$ is parallel to $z$. However, the following holds because of the bilinearity of the inner product: $$ \langle x,y \rangle=\langle x,z \rangle \Leftrightarrow \langle x,y-z \rangle=0 $$ which means that $x \perp y-z$.