## Finding the intersection of 2 lines in space

A common procedure used both in mathematics as well as computer science, involves the process of finding the intersection between 2 lines. 2 lines on a plane are fairly straightforward. Finding an intersection in space or in 3-dimensions has a few more steps, but can be just as easy.

Given 2 parametric equations:

**l1 & l2**

** x = 1 + 2t —— x = 0 + 5s **

**y = 0 + 3t —— y = 5 + 1s **

**z = 0 + 1t —— z = 5 – 3s**

Setting the **x**‘s and **y**‘s equal, we have to solve

**x: 1 + 2t = 0 + 5s & y: 0 + 3t = 5 + 1s**

Simplified to:

**x: 2t – 5s = -1 & **

**y: 3t – 1s = 5 **

Now to solve for one variable, the other must be cancelled.

A possible solution to this would be to multiply the first equation by -1 and the second equation by 5 resulting in:

**x: -2t + 5s = 1 **

**y: 15t – 5s = 25 **

From here the two **s** variables are equally opposite so we can add these two to cancel **s**.

Result

**13t = 26 => t = 2 **

Having **t** allows us to find **s** easily.

Simply replace **t** with 2 in the original equation.

**x: 1 + 2t = 0 + 5s**

**=> 1 + 2(2) = 5s => 5 = 5s => 1 = s**

Having **t** & **s**, you now have when the two line intercect. To ensure that the lines truly intersect, it is important to test these finding for all equations.

**x = 1 + 2(****2) = 5 x = 0 + 5(1)**** = 5 **

** y = 0 + 3(****2) = 6 y = 5 + 1****(1) = 6 **

** z = 0 + 1(****2) = 2 z = 5 – 3****(1) = 2 **

We are left with these same answers for both lines, so it can be concluded that these two lines intersect in space at **point(5, 6, 2)**.