Why can't I get this with Gram-Schmidt Can someone assist me with a very simple problem. I cannot get these two vectors to be orthogonal using Gram-Schmidt: 

$\{(1,-1,1),(2,1,1)\}$

What am I doing wrong?

Let $v_1=(1,-1,1)$.
$v_2 =
> (2,1,1)-\frac{\langle(2,1,1),(1,-1,1)\rangle}{\lvert (1,-1,1) \rvert^2}(1,-1,1)$  =$(2,1,1)-\frac{1}{3}(1,-1,1)$ This gives me
  $(\frac{5}{3},\frac{4}{3},-\frac{2}{3})$ Which is not orthogonal to
  $(1,-1,1)$.


The numerator of the fraction should be 
$$
\langle(2,1,1),(1,-1,1)\rangle
$$
Edit
After your correction, I can see that there is still an error, given that
$$
\frac{\langle(2,1,1),(1,-1,1)\rangle}{|(1,-1,1)|^2}=\frac{2}{3}
$$
