# Thread: Intersection of two planes

1. ## Intersection of two planes

I'm given the planes $x + y + z = 18$ and $4x + 3y - z = -3$. With these planes I am asked to find a parametric equation of the line in which these two planes intersect.

Then, I am asked to find the distance between this line and the point $(-5, 10, 13).$

2. Originally Posted by Ares_D1
I'm given the planes $x + y + z = 18$ and $4x + 3y - z = -3$. With these planes I am asked to find a parametric equation of the line in which these two planes intersect.

Then, I am asked to find the distance between this line and the point $(-5, 10, 13).$

Hi

You have several ways to find a parametric equation of the line of intersection

You can find 2 points on the line
Take for instance x=3, solve the system
$y + z = 15$
$3y - z = -15$

Then take x=-1, solve the system
$y + z = 19$
$3y - z = 1$

These 2 points define one direction vector of the line
Since you already have 1 point of the line, you can get a parametric equation

Second method, solve the system
$x + y + z = 18$
$4x + 3y - z = -3$
using z (for instance) as parameter

You will find x, y (and z) function of z (parameter)

3. Originally Posted by running-gag
Hi

You have several ways to find a parametric equation of the line of intersection

You can find 2 points on the line
Take for instance x=3, solve the system
$y + z = 15$
$3y - z = -15$

Then take x=-1, solve the system
$y + z = 19$
$3y - z = 1$

These 2 points define one direction vector of the line
Since you already have 1 point of the line, you can get a parametric equation

Second method, solve the system
$x + y + z = 18$
$4x + 3y - z = -3$
using z (for instance) as parameter

You will find x, y (and z) function of z (parameter)
Would solving for each variable by Gaussian elimination work in this case?

4. The system
$x + y + z = 18$
$4x + 3y - z = -3$

is a linear system with 2 equations and 3 unknowns
You can calculate each of the unknowns with respect to only one
You can use elimination if you want