Orthonormal basis for the orthogonal complement U $\subseteq$ $\mathbb R$$^5$ with span(v$_1$, v$_2$) and 
v$_1$=  $\begin{pmatrix}
    1 \\
    2 \\
    0 \\
    2 \\
    1 \\
    \end{pmatrix}$
 and v$_2$=  $\begin{pmatrix}
    1 \\
    1 \\
    1 \\
    1 \\
    1 \\
    \end{pmatrix}$
Now I want to find an orthonormal basis for U$^{\bot}$.
The following is the "theoretical part", please verify that my thoughts on that are right.
Because $\mathbb R$$^5$ = U $\oplus$ U$^{\bot}$ the dimension of U$^{\bot}$ = 3.
I need the vectors b$_1$, b$_2$ and b$_3$ in the span of U$^{\bot}$ and all of them need to fulfill that $\langle$v$_1$,b$_i$$\rangle$ = $0$ and $\langle$v$_2$, b$_i$$\rangle$ = $0$ (and of course they are linear independent).
After I find my 3 vectors b$_i$ I need to set them all to $\Vert$b$_i$$\Vert$ = 1 with $\hat b_i$ = b$_i$ * 1/$\Vert$b$_i$$\Vert$.
Now $\hat b_1$,$\hat b_2$ and $\hat b_3$ is the orthonormal basis of U$^{\bot}$. 
My problem is I don't know how to work with this "theoretical part" to find those b$_1$, b$_2$ and b$_3$.

By inspection we can easily choose


*

*$b_1=(1,0,0,0,-1)$

*$b_2=(0,1,0,-1,0)$

*$b_3=(2,-1,-1,0,0)$


such that $b_i\cdot v_j=0$ then we can orthonormalize them by GS process.
Otherwise we can use the vectors $e_i$ for the standard basis together with $v_1$ and $v_2$ as row vectors and obtain a basis by the RREF. Then we can orthonormalize that basis by GS process.
