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

 

[blue_raver22]

To find the intersection of two planes in R3, you can follow these steps:

1. Find the normal vectors of both planes: The normal vectors of the planes are the coefficients of the x, y, and z terms in the plane's equation. For example, if the equation of one plane is 2x + 3y - z = 6, then the normal vector would be (2, 3, -1).

2. Use the cross product to find the direction vector: The direction vector of the intersection line will be perpendicular to both normal vectors. To find it, take the cross product of the two normal vectors.

3. Find a point on the line: To find a point that lies on both planes, you can set one of the variables (x, y, or z) to 0 in one of the plane's equations and solve for the other two variables. This will give you a point that lies on both planes.

4. Write the equation of the line: Once you have the direction vector and a point on the line, you can use the parametric form of a line (x = x0 + at, y = y0 + bt, z = z0 + ct) to write the equation of the intersection line.

Alternatively, you can also use the symmetric form of a line ((x-x0)/a = (y-y0)/b = (z-z0)/c) to write the equation of the line, where (x0, y0, z0) is the point on the line and (a, b, c) is the direction vector.

I hope this helps! Let me know if you have any further questions.