Angle between two vectors after an orthogonal map

Two vectors $x,y\in\mathbb{R}^n$ make an angle of $75^\circ$. If $A$ is any $n\times n$ orthogonal matrix, what angle (in degrees) do $Ax$ and $Ay$ make?

$75^\circ$. Since $A^TA=I$, we have $(Ax)\cdot(Ay)=x^TA^TAy=x\cdot y$ and $\|Ax\|=\|x\|$, $\|Ay\|=\|y\|$, so the cosine of the angle is unchanged: orthogonal maps preserve inner products and hence angles.