I'm given the scalar product in $\mathbb{R}^3$ defined as $$\langle (x_1,x_2,x_3), (x_1,x_2,x_3)\rangle=x_1^2+2x_2x_1+2x_2^2+x_3^2$$
Let $B=\{ v_1,v_2,v_3\}$ a basis of $\mathbb{R}^3$ where $v_1=(-1,1,0)$, $v_2=(0,1,0)$, $v_3=(1,0,1)$. Applying the Gram Schmidt theorem to $B$ yields an orthogonal basis $\{ u_1,u_2,u_3\}$. I'm asked to find the coordinates of $u_3$ in the canonical basis, and I'm getting $$u_3=\Big(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\Big)$$, yet the correct answer supposedly is $$u_3=\Big( 0,-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\Big)$$I've checked multiple times yet I keep getting the same result and I'm starting to wonder if the teacher made a mistake. Is the teacher's answer correct?