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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$\begingroup$ Do you know of any ways to measure the angle between the vector $A$ and the vector going from the origin to $P$? $\endgroup$– Jesse MadnickDec 6, 2010 at 9:06
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$\begingroup$ @Jesse yes but only absolute value of it. $\endgroup$– MigolDec 6, 2010 at 9:10
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$\begingroup$ 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? $\endgroup$– Fredrik MeyerDec 6, 2010 at 9:21
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$\begingroup$ @Fredrik Yes, but in my case this formula is not helpful. Note that $cos( \theta ) = cos(0 - \theta )$. $\endgroup$– MigolDec 6, 2010 at 9:34
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$\begingroup$ 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 $\phi(P)$ $\endgroup$– J. J.Dec 6, 2010 at 9:59
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2 Answers
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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.
answered Dec 6, 2010 at 11:33