find an orthogonal basis for the space of solutions of the following equations

Question

Q.find an orthogonal basis for the space of solutions of the following equations 2x+y-z=0 y+z=0

How select solution? And how solve this question? ?

Answer 1

As I see, you have a intersection of two plans which is the solution of $\{2x+y-z=0,~y+z=0\}$. First solve it to find the solution: $$X= \left( \begin{array}{ccc} z \\ -z \\ z \end{array} \right)=z\left( \begin{array}{ccc} 1 \\ -1 \\ 1 \end{array} \right)$$ Now you have a requested vector.