# Line of intersection of Two Planes

1. Feb 20, 2013

### ChiralWaltz

1. The problem statement, all variables and given/known data

Please disregard, sign error corrected in the cross product

Determiner the line of intersection of the following two planes. Write the parametric equations for this line.

2x+y-2z=5
3x-6y-2z=15

2. Relevant equations

3. The attempt at a solution
First I crossed my normal vectors from the given equations: (2,1,-2)cross(3,-6,-2)= (-14,-2,15)

Then I solved my simultaneous equations starting with y=0, which gave me x=10 and z= (15/2).

To get into the parametric equation I took the point (10,0,15/2) + t(-14,-2,15), which gave me
x=10-14t
y=-2t
z=15t+(15/2)

Viewing this data in calcplot 3d shows that my line of intersection is no where near my planes.

Last edited: Feb 20, 2013
2. Feb 20, 2013

### ChiralWaltz

Error found in cross product (-14,-2,-15). Problem solved.

3. Feb 21, 2013

### HallsofIvy

Staff Emeritus
I'm not sure why you took the cross product of the normal vectors. Simply subtracting the first equation from the second eliminates z and gives x- 7y= 5.
That is, x= 7y+ 5 and putting that back into the first equation, 2x+y-2z= 2(7y+ 5)+ y- 2z= 15y+ 10- 2z= 5. 2z= 15y+ 5.

Parametric equations for the line of intersection are x= 7t+ 5, y= t, z= (15t+ 5)/2.