Interseccion of two planes in R3

[Kiwiro0ls]

How do you find the intersection of two planes in R3? The direction vector would be the cross product between the two normal vectors I imagine. So, how do I go about finding a point that lies in both planes so I can find the equation of the line?

Thanks :)

 

[Studiot]

What do you know about the equation of a plane in R³?

 

[Kiwiro0ls]

A plane is defined by a normal vector and a point. It can be written as
ax+by+cz=d where (a,b,c) is the normal vector and d is <(x1,y1,z1),(a,b,c)>

 

[Studiot]

It can be written as
ax+by+cz=d

Exactly so.

The general linear form in 3D is a plane, not a line.

In fact there is no single "equation of a line in 3D", which is probably why you can't find one.

Two planes intersect in a line so a line is defined by two planes.

A line has to be defined by two equations, not one.

a₁x+b₁y+c₁z=d₁ = P₁
a₂x+b₂y+c₂z=d₂ = P₂

For an alternative pair of equations see here

https://www.physicsforums.com/showthread.php?t=641057

However any linear combination of P₁ and P₂ will also contain this line. This can be described by the parameter λ such that

P₁ + λ(P₂) = 0

Edit:
This is referred to as a pencil of planes or a fan of planes or a sheaf of planes.

Wolfram have a good picture.
http://mathworld.wolfram.com/SheafofPlanes.html