Associated Topics || Dr. Math Home || Search Dr. Math

### 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
```
Associated Topics:
College Linear Algebra
High School Linear Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search