Sets of feasible directions

Question

I'm not exactly sure how the different points matter. I believe $p=[1,1,-1]^T, [2,-1,0]^T, [3,0,-1]^T, [0,3,-2]^T$ are all feasible directions.

Answer 1

Hint Feasible directions are a set of vectors along which you can move from some point $x$ staying feasible. Let's do one by example: take $x_a = (0,0,2)^T$ and notice that $x_a$ satisfies the equality constraint and the inequality constraints, so it is itself feasible. To keep feasible, you cannot decrease $x_1$ or $x_2$ components (lest the inequality constraints become violated) and you must stay on the plane $x_1 + 2x_2 + 3x_3 = 6$, so you cannot increase those components, without decreasing the third one, and the third one cannot increase by itself either. So whichever form your feasible directions take, they must look like $(a,b,c)$ with $c<0$ and $a,b>0$. Can you be more specific (they gotta satisfy a certain equality, you should be able to derive a 2D basis for them).