More on the Intersection of Two Lines in Three Dimensions

Date: 02/21/2009 at 09:00:33
From: Josemi
Subject: Intersection point of Two Lines

Hi,

Another Dr. Math conversation shows a very interesting formula for 
calculating the intersection point of two lines:

  a(V1 x V2) = (P2 - P1) x V2

(From http://mathforum.org/library/drmath/view/63719.html )

Would you please send me the steps to obtain "a"?

Your sincerely,
Josemi

Date: 02/23/2009 at 08:15:00
From: Doctor George
Subject: Re: Intersection point of Two Lines

Hi Josemi,

Look carefully at

  a(V1 x V2) = (P2 - P1) x V2

This is a vector equation.  For there to be a solution, the i,j,k 
components of each side must be equal.  So you could compare the i 
components to find the value of 'a' and then make sure this value 
also works for the j and k components.

If you are writing computer code, I would suggest this.  Take the dot 
product of both sides with V1 X V2 to obtain

  a (V1 x V2).(V1 x V2) = [(P2 - P1) x V2].(V1 x V2)

or

  a |V1 x V2)|^2 = [(P2 - P1) x V2].(V1 x V2)

This is now a scalar equation.  Just divide to solve for "a" on the 
left hand side.

Now, there is one last issue.  The whole procedure assumes that the 
two lines intersect.  The value we have for 'a' specifies a point on 
L1.  For two lines in 3D, we must make sure that this point is also a 
point on L2.  One way to do that is to check that the distance from 
this point to L2 is zero.

Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/