I have a vector A and a point P. My problem is how to find out if point is on the left or right side of vector looking from the point of origin of a vector in direction of it.
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Let $B$ be the vector from the end point of $A$ to $P$, then observe the sign of the coefficient of $A \wedge B$ in a chosen basis. The exact sign is up to $O(2)/SO(2)$, but you only need to distinguish between the two cases, so it's ok :)
UPD: Now that I think about it, letting $B$ be the vector from the starting point of $A$ to $P$ is even neater.