# Parametric Equation of a Line from the intersection of two planes

## Homework Statement

Find the parametric equation for a line of intersection of these two planes

x+2y+3z=0
4x+5y+6z=5

## Homework Equations

Normal to plane 1= <1,2,3>
Normal to plane 2= <4,5,6>

## The Attempt at a Solution

I know the way to do this problem is to take cross product of two normals etc etc,
but i want to know if the way i did this is correct also.

I already turned the work in so theres nothing i can do to change it but the curiosity is killing me.

First, i set the two planes equal to each other

4x+5y+6z-5=0
x+2y+3z=0

=> x+2y+3z=4x+5y+6z-5 (please correct me my thinking process is wrong,im winging it)
=> 3x+3y+3z=5
=> x+y+z=5/3 (now im thinking this is the equation of the intersection of the two planes but this isnt the equation of a line, it looks like a plane, or is it?)

so i took a point on this set, (5/3,0,0) and two other points (0,5/3,0) and (0,0,5/3)

i did <0,0,5/3> - <0,5/3,0> = <0,-5/3,5/3> as a directional vector.

so L(t)= (5/3,0,0) + <0,-5/3,5/3>t

x=5/3
y=-5/3t
z=5/3t

IS any of this wrong?!

Dick
Homework Helper
Try it out. Put your x,y and z into the two original plane equations. At t=0 you get (5/3,0,0). That's not on either of the original planes, is it?

Ah i do notice that, but

x+2y+3z=4x+5y+6z-5

if i plug them into that those points solve the equation, which is the x,y,z such that those two planes are equal, or is that me failing at winging a problem?

Dick
Homework Helper
If you mix up two planes you get a third plane. A line in that plane may or may not be in either of the two planes you started with. I admire your spirit of winging it, but when you scamble the two planes, you loose information.

so i just needed to restrict that plane to a set of values that lie in both planes. so im just wondering what exactly did i do by setting the two planes equal to each other? i notice i get the same result if i subtract equation 1 from equation 2, so did i just subtract the two planes by setting them equal? im confused as to what that actually does

vela
Staff Emeritus
Homework Helper
Yeah, you effectively just subtracted one equation from the other. This gives you a new plane in which the line of intersection lies, so you're just finding the equation of a plane rotated about that line.

Dick