$A$ is an $n\times n$ orthogonal matrix. Show that $(A\mathbf x)\cdot (A \mathbf y ) = \mathbf x\cdot\mathbf y$ for all $\mathbf x$ and $\mathbf y\in \mathbb{R}^n$.
I wasn't sure how to treat the $\mathbf x$ and $\mathbf y$ terms, are they just the $(x_1, x_2, ... x_n)$ column vectors?
Also, I tried one thing but it seemed like it was too simple and I couldn't be sure if it was the correct procedure to show the equality is true or not; however, this is what I did:
$(A\mathbf x)^T(A\mathbf x)~\cdot(A\mathbf y) = (A\mathbf x)^T(\mathbf x\cdot\mathbf y)$
$I\cdot (A\mathbf y) = AI\mathbf y$
$A\mathbf y= A\mathbf y$
Is this the correct path, I don't feel like I am understand everything that is going on with this problem. If this is the correct method, could someone please explain what is happening on a deeper level?