Finding the standard equation for a plane orthogonal to two other planes

[Guidenable]

Homework Statement

let p1 and p2 be planes in R3, with respective equations:

x+5y-z=20 and 2x+5y+2z=20

These planes are not parallel. Find the standard equation for the plane that is orthogonal to both of these planes and contains the origin.

The Attempt at a Solution

I have only managed to garner a few facts from the problem, however I don't know how to use them. Here they are:

Since the plane, we'll call it ζ that we are looking for is orthogonal to both of these planes, ζ must contain the normal vectors of both of these planes. These normal vectors, for p1 and p2 respectively, are (1,5,-1) and (2,5,2). Also, the standard equation of ζ must be equal to 0, as ζ contains the origin, ie:

ax+by+cz=0, since the origin is (x,y,z)=(0,0,0)

That's as far as I got. The information is there, I just have no clue how to use it.

 

[Kreizhn]

Well, you've said that your plane must contain the normal vectors (x,y,z) = (1,5,-1) and (2,5,2) and looks like ax+by+cz = 0. Maybe you could put those vectors into the formula for the plane and come up with some equations...

 

[Guidenable]

Oh wow. Thanks a bunch, can't believe I didn't see that XD