Calculation on inner product space.

Question

$V$ : vector space on $\mathbb{R}$ or $\mathbb{C}$

$ 〈 \cdot , \cdot 〉 $ : inner product on V

$x,y,z \in V$

Proposition;

If $y\neq 0$ and $z=\dfrac{〈x,y〉}{〈y,y〉} y$ hold, then $〈z,z〉=〈x,z〉$ holds.

Why this proposition holds? I tried to calculate

\begin{equation} 〈z,z〉=〈z,\dfrac{〈x,y〉}{〈y,y〉} y〉=\dfrac{〈x,y〉}{〈y,y〉} 〈z,y〉=\cdots \end{equation}

I cannot lead to the conclusion.

Answer 1

Let $c=\langle x, y \rangle$ and $d=\langle y, y \rangle$ The $\langle z, z \rangle =\langle (cy/d), (cy/d) \rangle =\frac {|c|^{2}} {d^{2}} \langle y, y \rangle= \frac {|c|^{2}} {d} $.

Also $\langle x, z \rangle =\frac {\overline {c}} d \langle x, y \rangle= \frac {|c|^{2}} {d} $.