Let $V$ be an inner product space. Let$(v_1,v_2,...)$ be a sequence of linearly independent vectors in $V$ and let$$a_1=\frac{v_1}{\|v_1\|},a_{n+1}=\frac{v_{n+1}-\langle v_{n+1},a_1\rangle a_1-...-\langle v_{n+1},a_n\rangle a_n}{\|v_{n+1}-\langle v_{n+1},a_1\rangle a_1-...-\langle v_{n+1},a_n\rangle a_n\|}$$Show that $\|v_{n+1}-\langle v_{n+1},a_1\rangle a_1-...-\langle v_{n+1},a_n\rangle a_n\|=det(A_n)$ where $A_n=(\langle v_1,v_j\rangle )_{1\le i,j\le n}$
I already proved that $(a_1,a_2,...)$ is a orthonormal sequence and I am wondering how to deal with $\det(A_n)$.