I have a set $W=span${$x\_1,x\_2$}, where $x\_1=(1,3,-3)$ and $x\_2=(5,0,3)$. I am told that the set {$v\_1,v\_2$} is an orthogonal basis for the set $W$, and that $v\_1=x\_1=(1,3,-3)$, and I need to find $v\_2$.

I started off by letting $v\_2=(a,b,c)$. I need to find a system of three linear equations which I can solve for $a,b$ and $c$. Since the set is orthogonal, I know that the dot product of $v\_1$ and $v\_2$ is equal to zero which gives the first equation:

$$(1,3,-3)\cdot(a,b,c)=0$$
$$a+3b-3c=0$$

However I do not know what else I can do to find two other equations. Any help would be appreciated