I have an exercise that says: Given $Q(x) = 3x_1^2+8x_1x_2+9x_2^2$ find a unit vector $x$ that minimizes $Q(x)$ subject to constraint $x^Tx=1$.
I set up $A=$$\begin{bmatrix}3&4\\4&9\end{bmatrix}$ and found the eigenvalues of this matrix to be $λ_1=11$ and $λ_2=1$ and the corresponding unit eigenvectors $u_1=$$\begin{bmatrix}\frac{1}{\sqrt 5}\\\frac{2}{\sqrt 5}\end{bmatrix}$ and $u_2=$$\begin{bmatrix}\frac{-2}{\sqrt 5}\\\frac{1}{\sqrt 5}\end{bmatrix}$. I then applied a transformation $x=Py$ and get the new equation $Q(y)=11y_1^2+y_2^2$ which my professor just rewrites as $Q(x)=11x_1^2+x_2^2$ (to match the x variable notation used by the constraints I guess?). Anyways, to get the minimum, I need to add the constraint $x^Tu_1=0$, which according to my textbook results in $Q(x)=x_2^2$.
My Question: How does $x^Tu_1=0$ for the constraint? If I plug in $u_1$ to that equation there is no way I can get $x^T$ which results in $x^Tu_1=0$ to get the equation $Q(x)=x_2^2$. (I know the minimize procedure can be done without using $x^Tu_1=0$, but I wanted to do it this way to show the question I have).
Note: I apologize if the wording of this question is a little messy/scattered. I didn't really know how to explain it better than this. I think the issue might be how the notation is presented in the textbook I'm using but I'm not sure. Many thanks to anyone who can help clear things up for me here.