Question: Find an orthonormal basis $u_1$,$u_2$,$u_3$ for $\mathbb{C}^3$ such that $u_1$ is a multiple of $(1,w,w^2)$, where $w=e^{2i\pi/3}$.
I know that I must apply Gram-Schmidt, but I am unsure how to even get $u_1$. I don't know how to work out norms when there is an exponential.
To get $u_1$, divide $(1,w,w^2)$ by its norm. The norm is $$\sqrt{1+|w|^2+|w^2|^2}=\sqrt{1+1+1}=\sqrt{3}.$$ This is because $w=e^{i2\pi/3}$ has absolute value $1$. The norm of $re^{i\theta}$ is $r$.