Use Gram-Schmidt to replace set S by an Orthogonal set with the same span

Question

S = $ \begin{Bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix} \begin{bmatrix} 5 \\ -1 \\ 2 \\ \end{bmatrix} \end{Bmatrix} $

So after doing some work I get:

$v_1 = $ $ \begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix} $

$v_2 = 1/5$ $ \begin{bmatrix} 24 \\ -6 \\ 9 \\ \end{bmatrix} $

But the answer is the following:

$ \begin{Bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix} \begin{bmatrix} 3 \\ -3 \\ 0 \\ \end{bmatrix} \end{Bmatrix} $

So is my $v_2$ wrong? Am I missing a step?

Answer 1

Perhaps here is the mistake:

$$u_1=\frac{v_1}{\left\|v_1\right\|}=\frac1{\sqrt3}\begin{pmatrix}1\\1\\1\end{pmatrix}$$

and now

$$v_2-(v_2\cdot u_1)u_1=\begin{pmatrix}5\\\!-1\\2\end{pmatrix}-\frac136\begin{pmatrix}1\\1\\1\end{pmatrix}=\begin{pmatrix}3\\\!-3\\0\end{pmatrix}$$