Finding four linearly independent vectors orthogonal to each other

Question

I came up with these vectors:

u = \begin{bmatrix}1\\ 1\\ 0 \\1 \\ \end{bmatrix} v = \begin{bmatrix}1\\ -1\\ 1 \\0 \\ \end{bmatrix} w = \begin{bmatrix}2\\ -2\\ -4 \\0 \\ \end{bmatrix}

u and v are orthogonal. w is orthogonal to u and v. I have to find another matrix z that's orthogonal to u, v, and w, and I'm having a lot of trouble. Is there a better or more efficient way to do this than to simply keep guessing random vectors and adjusting them?

Answer 1

In four dimensional euclidean space, the standard unit vectors $(1,0,0,0)\,,(0,1,0,0)\,,(0,0,1,0)$ and $(0,0,0,1)$ satisfy your condition.