Matrix with orthonormal base I have the two following given vectors:
$\vec{v_{1} }=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$
 $\vec{v_{2} }=\begin{pmatrix} 3 \\ 0 \\ -3 \end{pmatrix}  $
I have to calculate matrix $B$ so that these vectors in $\mathbb{R}^{3}$ construct an orthonormal basis.
The solution is: 
$$B=\begin{pmatrix} 0 & -\frac{\sqrt{2} }{2}  & \frac{\sqrt{2} }{2}  \\ 1 & 0 & 0 \\ 0 & -\frac{\sqrt{2} }{2}  & -\frac{\sqrt{2} }{2}   \end{pmatrix}$$
I really don't have any idea how to get this matrix. I'm also confused because I only have 2 vectors.

The 3rd vector can be cross product of the given two vector $v_1$ and $v_2$.
$$v_3=v_1 \times v_2=(\begin{array}-3 & 0 & -3 \end{array})$$
You already know that  $v_1$ and $v_2$ are orthogonal. So the rest to do is to normalize the vectors. 
