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