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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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Do you know of any ways to measure the angle between the vector $A$ and the vector going from the origin to $P$? – Jesse Madnick Dec 6 '10 at 9:06
@Jesse yes but only absolute value of it. – Migol Dec 6 '10 at 9:10
Do you know the formula $a \cdot b = |a||b|\cos \theta$ (where $\theta$ is the angle between the vectors)? In case the answer is yes, how can you use this formula to answer your question? – Fredrik Meyer Dec 6 '10 at 9:21
@Fredrik Yes, but in my case this formula is not helpful. Note that $cos( \theta ) = cos(0 - \theta )$. – Migol Dec 6 '10 at 9:34
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 $\phi(P)$ – J. J. Dec 6 '10 at 9:59

2 Answers

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This answer on MO should do it for you.

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up vote 0 down vote

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.

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