Write the set of vectors that are orthogonal to $v$ as a linear combination of two unit vectors.

Question

$$v = \langle 1, -\sqrt 8, -\sqrt 8\rangle \text{ is a vector.}$$

I know I have to find two unit vectors $u$ and $w$ so that any vector that is orthogonal to $v$ can be expressed as a linear combination of $u$ and $w$.

But after that, I'm lost. Help, please.

Answer 1

Make an ansatz $u=\langle x,y,0\rangle $ and from $v\perp u $ and $\|u\|=1$ obtain conditions for $x,y$. After that, do the same with $w=\langle 0,y,z\rangle$. (Apparently you are not asked to additionally ensure $u\perp w$)