# Intersecting planes

1. Nov 20, 2005

### cscott

What's the easiest way to make up two equations of a plane that intersect at a line?

2. Nov 20, 2005

### TD

What are you looking for, the equation of the line of intersection? Because "two equations of a plane" doesn't really make sense to me...

3. Nov 20, 2005

### cscott

I need two equations (Ax + By + Cz + D = 0) that describe two planes who's intersection point is a line.

Last edited: Nov 20, 2005
4. Nov 20, 2005

### vaishakh

Lines which are not parallel intersect with each other. so don't make them parallel. lines are parallel if their slopes are same. so think what have you to change or not to? A, B, C or D

5. Nov 20, 2005

### cscott

I understand what you're saying but I'm not sure how the slope is represented in that form for 3-dimensions.

6. Nov 21, 2005

### HallsofIvy

Staff Emeritus
How do you generally write the equation of a plane?

7. Nov 21, 2005

### TD

I'm still not 100% sure what you mean but I *think* you mean that you're looking for the equations of two planes which, together as a system, form the equation of their intersection line (assuming the planes weren't parallel). Is this correct?

8. Nov 21, 2005

### cscott

That is correct! How do I make sure they aren't parallel?

9. Nov 21, 2005

### fomenkoa

Scott: This is pretty easy!

The only thing u do is make sure both bormal vectors are not scalar multiples of eachother..... in other words

if P1 = Ax+By+Cz+D
and P2 = Wx + Xy +Yz +Z

then to intesect in a line..... [A,B,C] canot equal k[W,X,Y] k is any num

Anton

10. Nov 21, 2005

### cscott

Alright, thanks.

11. Nov 21, 2005

### TD

Indeed, so with two planes

$$\begin{array}{l} \alpha :ax + by + cz + d = 0 \\ \beta :a'x + b'y + c'z + d' = 0 \\ \end{array}$$

the line of intersection is given by

$$\left\{ \begin{array}{l} ax + by + cz + d = 0 \\ a'x + b'y + c'z + d' = 0 \\ \end{array} \right$$

under the condition that

$$\alpha \not\parallel \beta \Leftrightarrow \left( {a,b,c} \right) \ne k\left( {a',b',c'} \right)\forall k \in \mathbb{Z}$$