Consider the cone $K$ in $\mathbb{R}^{3}$ defined by

$$x_{3}^{2}=x_{1}^{2}+x_{2}^{2}, \quad x_{3}>0 .$$

Find a unit normal $\mathbf{n}=\left(n_{1}, n_{2}, n_{3}\right)$ to $K$ at the point $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$ such that $n_{3} \geqslant 0$.

Show that if $\mathbf{p}=\left(p_{1}, p_{2}, p_{3}\right)$ satisfies

$$p_{3}^{2} \geqslant p_{1}^{2}+p_{2}^{2}$$

and $p_{3} \geqslant 0$ then

$$\mathbf{p} \cdot \mathbf{n} \geqslant 0$$