I know one way to do it. We want the point with maximal projection over a fixed vector. Let's treat points as vectors starting at $$$(0,0)$$$. We know that $$$A \cdot B = |A|\cdot|B|\cdot \cos{\theta}$$$, where $$$\theta$$$ is the angle between $$$A$$$ and $$$B$$$. Well, if $$$B$$$ was the $$$X$$$-axis, then $$$|A|\cdot \cos \theta$$$ is the $$$X$$$-component in this new system, so $$$A \cdot B$$$ is the projection of $$$A$$$ over $$$B$$$, scaled by $$$|B|$$$. Therefore, the point with maximal projection over $$$B$$$ is the point with maximal dot product with $$$B$$$, since all projection are scaled by some constant amount -$$$|B|$$$. To find that point we can maintain the upper hull of the points increasingly sorted by $$$X$$$ and do binary search. To find the point with minimal projection over a fixed vector we can do the same but maintaining the Lower hull

But I've read in some article that there are a few ways to do it, so I want to know those ways.