[Math] Set of Feasible Directions

Question

I don't even know what to do for the first part. How do you even find all the feasible directions of a particular Set…?

Then how do you proceed to finding basic directions?

Answer 1

Use the definition of a feasible direction $d$ at $(0,0,1)$: $(0,0,1)+\theta d\in P$ for all $\theta\in (0,\delta)$ for some $\delta>0$ if and only if $\theta d_1 +\theta d_2 +1+\theta d_3=1$ and $\theta d_1\geq 0$, $\theta d_2\geq 0$ and $1+\theta d_3\geq 0$ for all $\theta\in (0,\delta)$. Since $\theta\in (0,\delta)$, this is equivalent to $$d_1+d_2+d_3=0,\quad d_1,d_2\geq 0,\quad \theta d_3\geq -1$$ for all $\theta\in (0,\delta)$. This defines the feasible directions.