I am pretty sure I am missing something somewhere but I cannot see where and have been stuck for a while... Like in the example I set the first vector $U_1$ to \begin{pmatrix} -1\\ 1\\ 0\\ \end{pmatrix} and then, its orthogonal vector, $U_2$ as $x_1 - \frac{U_1^T U_1}{||U_1||} x_1$.
My issue is evaluating the second term, in that I get a vector that is not orthogonal to \begin{pmatrix} -1\\ 1\\ 0\\ \end{pmatrix}
I am attempting to show my working here: $U_2 =$ $$ \begin{pmatrix} -1\\ 0\\ 1\\ \end{pmatrix} - {{2}^{1/2}}\begin{pmatrix} -1\\ 0\\ 1\\ \end{pmatrix} $$
This gives me $$\begin{pmatrix} -1 + {2}^{1/2}\\ 0\\ 1- {2}^{1/2}\\ \end{pmatrix}$$ which when I take the dot product with my original $U_1$, does not give 0 and is hence not orthogonal. Apologies for the mess as I am not accustomed to the formatting. Would really appreciate some help.
