TITLE: How to determine the side on which a point lies?
QUESTION [10 upvotes]: Suppose we have a linear equation and a point in the plane, then how can one determine on which side of the line the point lies?

REPLY [10 votes]: Let your line be given by $ax+by=d$, and call $\vec n=(a,b)$ the normal vector of the line. Let's label the side $\vec n$ points to $+$ and the opposite side $-$. Then for any point $(x,y)$ in the plane, the sign of
$$ax+by-d$$
determines which side the point $(x,y)$ is on. Notice that this is $0$ if and only if $(x,y)$ is on the line, so all points not on the line get $+$ or $-$.
Here's a picture illustrating the situation: