Vector Algebra: Finding the Intersection Point
Date: 04/23/2003 at 06:56:56 From: Patricia Subject: Finding the intersection point of two lines in 3D If I have two lines in three dimensions that I know intersect at some point, how do I work out what that point is? Both lines are defined by two points on each line. Many thanks.
Date: 04/23/2003 at 09:54:29 From: Doctor George Subject: Re: Finding the intersection point of two lines in 3D Hi Patricia, Thanks for writing to Doctor Math. Let's try this with vector algebra. First write the two equations like this. L1 = P1 + a V1 L2 = P2 + b V2 P1 and P2 are points on each line. V1 and V2 are the direction vectors for each line. If we assume that the lines intersect, we can look for the point on L1 that satisfies the equation for L2. This gives us this equation to solve. P1 + a V1 = P2 + b V2 Now rewrite it like this. a V1 = (P2 - P1) + b V2 Now take the cross product of each side with V2. This will make the term with 'b' drop out. a (V1 X V2) = (P2 - P1) X V2 If the lines intersect at a single point, then the resultant vectors on each side of this equation must be parallel, and the left side must not be the zero vector. We should check to make sure that this is true. Once we have checked this, we can solve for 'a' by taking the magnitude of each side and dividing. If the resultant vectors are parallel, but in opposite directions, then 'a' is the negative of the ratio of magnitudes. Once we have 'a' we can go back to the equation for L1 to find the intersection point. Write back if you need more help with this. - Doctor George, The Math Forum http://mathforum.org/dr.math/
Date: 04/24/2003 at 08:28:05 From: Patricia Subject: Thank you (Finding the intersection point of two lines in 3D) Very many thanks for your help and quick response. Kind regards, Patricia
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