Let $v_1,...,v_n$ and $w_1,...w_n$ be two sets of linearly independent vectors in $\mathbb{R^n}$. Show that all their dot products are the same, so $v_j \dot\ v_i = w_i \dot\ w_j$ for all $i,j = 1,...,n$ iff there is an orthogonal matrix $Q$ such that $w_i = Qv_i$ for all $i=1,...,n$.
My attempt:
$\langle v+w,v+w\rangle = \langle v,v\rangle+\langle v,w\rangle+\langle w,v\rangle+\langle w,w\rangle$ and since $w_i = Qv_i$ we have that $\|v\|^2 = \|w\|^2$ ?