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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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    One way is to perform a rotation $\phi$ so that $\phi(A)$ lies on the x-axis and then inspect the sign of the y-coordinate of $\phi(P)$2010-12-06

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This answer on MO should do it for you.

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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.