Finding the dual cone of $\Omega_1=\{(x_1,x_2)\in\mathbb{R}^2:x_2\geq 0\}$

Question

I am trying to find the dual cone of $$\Omega_1=\{(x_1,x_2)\in\mathbb{R}^2:x_2\geq 0\}.$$ Let $K$ be a convex cone in an inner product space $H$. Then the dual cone,$\ K^*$, is defined by $$K^*=\{x\in H:\langle z,x\rangle\geq 0 \ \ \forall z\in K\}.$$ I am having difficulty visualising what the dual cone should look like, if the cone is the upper half-plane.