Find a vector that is perpendicular to two other 4 dimensional vectors

Question

I have the following problem:

"Given the two vectors $w_1=(5,0,1,1)$ and $w_2=(1,1,0,5)$.

Find a vector that is orthogonal to both $w_1$ and $w_2$."

I tried to create an augmented matrix, and reduce it to echelon form and this is what I got:

$$ F= \left(\begin{matrix}1&0&\frac{1}{5}&\frac{1}{5}&0\\0&1&\frac{-1}{5}&\frac{24}{5}&0\end{matrix}\right) $$

But how do I continue from this. I know I'm supposed to make a parametric form, but I don't know how to handle this matrix.

Answer 1

$(0,1,1)\times(1,0,5)=(5,1,-1)$ is perpendicular to $(0,1,1)$ and $(1,0,5)$,

so $(0,5,1,-1)$ is perpendicular to $(t,0,1,1)$ and $(u,1,0,5)$.

Answer 2

All you have to do is to find a solution of the system$$\left\{\begin{array}{l}5a+c+d=0\\a+b+5d=0.\end{array}\right.$$One such solution is $(0,0,0,0)$, of course. If you're after a non-zero solution, you can take $(1,-1,-5,0)$, for instance