Perpendicular vectors of equal length and orthogonal matrices

Question

I have an $n \times n$ orthogonal matrix $A$ and vectors $x,y \in \mathbb{R}^n$ and $\theta \in (0,\pi)$ and this satisfies

$Ax = \cos\theta x - \sin\theta y$, and $Ay = \sin\theta x + \cos\theta y$

I must now proof that $x$ and $y$ are perpendicular vectors of equal length.

I know this means $|x|^2 = |y|^2$ and $x\cdot y=0$, and I also think I should use the preserved Euclidean inner product. How to continue?

Answer 1

You're correct. In formulaic terms, the fact that $A$ preserves the Euclidean i.p. is just the property $$(Ax) \cdot (Ay) = x \cdot y .$$