# Equation of a plane perpendicular to 2 other planes

• XtremeThunder
In summary, the problem is to find the equation of a plane through the origin and perpendicular to the planes given by x-y+z=5 and 2x+y-2z=7. The first step is to find the normal vectors of each given plane. Then, the normal vector of the new plane must be perpendicular to both given planes, so it can be found by taking the cross product of the two normal vectors. Finally, the coefficients of the final equation can be determined using the result of the cross product as the normal vector.
XtremeThunder
Greetings, I am have a bit of a problem getting started on the following problem. The only thing that is confusing me is I am use to seeing a problem like this be perpendicular to 1 plane and not 2.

The problem I am working with is as follows:

Find the equation of a plane through the origin and perpendicular to the planes given by x-y+z=5 and 2x+y-2z=7.

I just need some direction on where to start.

Edit:
I am not quite sure what to do with both planes given.

The final equation should look similar to the following, correct?:
?(x-0)+?(y-0)+?(z-0)=0. Again, I am not sure how to come up with the coefficients using two given planes.

I spent about 4 hours on this problem and feel really lost and stupid!

-Joe

Last edited:
Are you familiar with the concept of normal vector ?

Yes, I am familiar with the concept of a normal vector.

Would I be correct in saying each plane's normal vector is i-j+k and 2i+j-2k, or is that totally off base?

Or would I use the cross product between the two?

Yes,exactly. If the new plane is perpendicular to both the given planes, its normal vector must be perpendicular to the normal vectors of both given planes- and thus parallel to the cross product of the two normal vectors

The coefficients of the equation of my first post (?(x-0)+?(y-0)+?(z-0)=0) would be the coefficients of the result of the cross product between the two normal vectors i-j+k and 2i+j-2k, correct, which would be the final answer?

## 1. What is the equation of a plane perpendicular to two other planes?

The equation of a plane perpendicular to two other planes can be found by taking the cross product of the normal vectors of the given planes. This will give the direction vector of the new plane, which can then be used to find the equation.

## 2. How do I determine if a plane is perpendicular to two other planes?

A plane is perpendicular to two other planes if its normal vector is perpendicular to the normal vectors of both planes. This means that the dot product of the normal vector of the new plane with the normal vectors of the given planes will be equal to zero.

## 3. Can a plane be perpendicular to two parallel planes?

No, if two planes are parallel, their normal vectors will be parallel as well. This means that the dot product of the normal vector of the new plane with the normal vectors of the given planes will not be equal to zero, and therefore the new plane will not be perpendicular to the given planes.

## 4. Can there be multiple planes perpendicular to two given planes?

Yes, there can be multiple planes perpendicular to two given planes. This is because there are infinite possible directions for the normal vector of the new plane, as long as it is perpendicular to the normal vectors of the given planes.

## 5. How can I find the distance between a point and a plane perpendicular to two other planes?

The distance between a point and a plane perpendicular to two other planes can be found by using the distance formula, where the point is the given point and the plane is defined by the equation found using the cross product of the normal vectors of the given planes. This distance formula is also known as the Hesse normal form of a plane.

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