Orthogonal Projection onto a Half Space

Question

How to find the projection(the closest point) of point $a$ to half-space $(p,x)\le\alpha$, where $p,x\in\mathbb R^n$,$\alpha \in \mathbb R$? $(p,x)$ is dot product.

Answer 1

1) Find a normal vector $\vec \nu$ to the hyperplane $\mathrm H = \{x|p.x = \alpha\}$ ;

2) Calculate the distance $d$ between $a$ and $\rm H$ ;

3) Then, then point you want is $a - d \vec \nu$.