Normal vector of surface

Question

Elementary question: If I am given a plane in $3$-space represented as $ax+by=c$, the normal vector should be $\vec{n}=<a, b, 0>,$ correct? Since $z$ attains any value in the plane, why do we assign it a value of $0$ in the normal vector expression? A better question, how can we assign it any value when it attains every value of $z$? Is it because $<a,b,0>$ is just one of the normal vectors to the plane?

Answer 1

$z$ attains all values, making the plane a vertical wall going in the $z-$direction. So any vector normal to this plane cannot have any vertical component, or else it's not normal.

$<a,b,0>$ is the only normal vector to the plane. Since the plane is linear, its direction does not change, so neither does its normal vector