Find $v_3$ such that $\{v_0, v_1, v_2, v_3\}$ is an orthogonal basis of $\mathbb{R}^4$?
let $v_3 = \left[\begin{array}{vector}
a\\
b\\
c\\
d\end{array}\right]$, then $&lt;v_0, v_3&gt;=0, &lt;v_1, v_3&gt;=0, &lt;v_2, v_3&gt;=0, &lt;v_3, v_3&gt;=1$,
reduce to 

a+b=0 
a+c=0
a-d=0


and $a^2+b^2+c^2+d^2 = 1$

so there are many solutions, one example can be 

$ \left[\begin{array}{arr}
 \frac{1}{2}\\
 -\frac{1}{2}\\
 -\frac{1}{2}\\
 \frac{1}{2}\end
{array} \right]$