How to prove these two vectors are orthogonal?

Question

Let $$Q = I - 2\frac{vv^T}{v^Tv}$$ And let $y$ be any vector orthogonal to $v$, and $x$ be any vector. Show that $x − Qx$ is orthogonal to $y$, i.e. $Q$ projects through any vector that is orthogonal to $v$.

I have done this so far, and don't know what to do next: $$\begin{align} &(x − Qx)y\\ &=[x - (I - 2\frac{vv^T}{v^Tv})x]y\\ &=x - xI + 2x\frac{vv^T}{v^Tv}y\\ &=2x\frac{vv^T}{v^Tv}y\\ &= ? \end{align}$$

I know I have to prove $(x − Qx)y=I$, but I son't know what to do next, when I reached here.

Answer 1

Show that $x−Qx$ is orthogonal to $y$

$\vdots$

I know I have to prove $(x − Qx)y=I$.

Wrong. $x-Qx$ is orthogonal to $y$ if $(x-Qx)^Ty=0$ and you are almost done. You still need to use that $y$ is orthogonal to $v$, which means $v^Ty=0$. This yields \begin{align} (x-Qx)^Ty=\ldots=2x^T\frac{vv^T}{v^Tv}y=\frac{2}{v^Tv}x^Tv\overbrace{v^Ty}^{=0}=0. \end{align}