Find the orthogonal complement $W^{\bot}$ of W and give a basis for $W^{\bot}$

Question

I have no clue how to start solving this. I've been watching videos and looking at examples but I can't find any. I couldn't find an example in my textbook either.

$$W= \begin{Bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}: 2x-y+z=0 \end{Bmatrix} $$

Can I get a few hints on how to go about this?

Answer 1

Note that $$W= \{ (x,y,z)^T : (2, -1, 1) \cdot (x,y,z)^T = 0\}$$ where $\cdot$ denotes the scalar product. So $W$ is exactly the orthogonal of the 1-dimensional space spanned by $(2, -1, 1)$.

Now, what can we say about the orthogonal of $W$?