Is $Av_1,Av_2,Av_3$ orthogonal if you have eigenvector of $A^TA$

Question

Let $A\in M_3(\mathbb R)$ and if $v_1,v_2,v_3$ orthonormed eigenvectors of matrix $A^TA$ and which eigenvalues is $1,2,3$ then vectors $Av_1,Av_2,Av_3$ is orthogonal?

I only know that we need to prove that $(Av_1,Av_2)=0$ and $(Av_1,Av_3)=0$ but I write that $(Av_1,Av_2)=(Av_1)^TAv_2=v_1^TA^TAv_2=2v_1^Tv_2=2(v_1,v_2)=0$ but I do not know is this prove ok?