How to write vectors in abbreviated set notation?

Question

I was wondering whether anyone knew how to write a vectors in abbreviated set notation to express the solutions to this question:

"Determine all values of x, y, z ∈ R such that (x, y, z) is perpendicular to both a = (1, 1, 1) and b = (−1, 1, 1)."

Letting n=[x , y, z], I figured out the two simultaneous equations (we have not covered cross product yet)

1. x + y + z = 0
2. -x + y + z = 0

However, the question wants us to express the answer in the form of {...|c ∈ R) which I am unsure how to do.

I understanding expressing the answer in the regular notation would be something like {(x, y , z) ∈ $R^3$ | x=0 and y=-z}.

Thank you very much for your help guys! Much appreciated :)

Answer 1

There are infinitely many vectors $(x,y,z)$ perpendicular to $(1,1,1)$ and $(-1,1,1)$;

if you find one of them, say ($l,m,n$), then all vectors in the set {$(cl,cm,cn)|c\in \mathbb R$} are solutions.

You found a solution $(0,1,-1),$ so {$(0,c,-c)|c\in\mathbb R$} are solutions.

Answer 2

The reduced (row) echelon form of the homogeneous linear system above (equations 1. and 2.) is $$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix}, $$ which tells us that $x = 0$ and $y = -z$. By convention, $z$ is a free variable, so every solution is of the form $$ \{ (0,-c,c) \in \mathbb{R}^3 \mid c \in \mathbb{R} \}.$$