# Equation of plane containing the intersection of two planes

• morsel
In summary, the conversation discusses finding the equation of a plane that passes through a given point and contains the line of intersection of two other planes. The formula for the equation of a plane is provided and the individual attempts at solving the problem are outlined. The final proposed equation is x - 2y + 4z = -1, which is believed to be correct but may need further verification.
morsel

## Homework Statement

Find an equation of the plane that passes through the point (-1,2,1) and contains the line of intersection of the planes x + y - z = 2 and 2x - y + 3z = 1

## Homework Equations

Equation of a plane:
a(x-x0) + b(y-y0) + c(z-z0) = 0

## The Attempt at a Solution

n1 = <1,1,-1>
n2 = <2,-1,3>
n1$$\times$$n2 = <2,-5,-3> = <a,b,c>

Equation:
2(x+1) - 5(y-2) - 3(z-1) = 0
2x - 5y - 3z = -15

What am I doing wrong? Any help is appreciated.

What makes you think you are doing anything wrong?

The solutions section in the back of my textbook says the answer is x - 2y + 4z = -1.
My textbook occasionally has wrong answers, though. So my answer is OK?

Looking at it quickly, it looks right to me. But if you want to be absolutely certain, you could determine a couple of points on the intersection of the planes by using their equations. Then if those two points are on your proposed plane the whole line of intersection must be and you are certain.

## What is the equation of a plane containing the intersection of two planes?

The equation of a plane containing the intersection of two planes can be found by taking the cross product of the normal vectors of the two planes. This will give you the coefficients of the x, y, and z terms in the equation.

## How do you find the normal vectors of two planes?

To find the normal vectors of two planes, you can take any two non-parallel vectors that lie in each plane and take their cross product. The resulting vector will be orthogonal to both planes and can be used as the normal vector for each plane.

## Can the intersection of two planes be a line?

Yes, it is possible for the intersection of two planes to be a line. This occurs when the two planes are parallel to each other.

## What happens if the two planes are identical?

If the two planes are identical, then their intersection will also be the same plane. In this case, the equation of the plane containing the intersection will be the same as the equation of the two given planes.

## Are there any other methods for finding the equation of a plane containing the intersection of two planes?

Yes, there are other methods such as using the scalar equation of a plane or using the distance formula to find the equation. However, taking the cross product of the normal vectors is the most common and efficient method.

• Calculus and Beyond Homework Help
Replies
1
Views
286
• Calculus and Beyond Homework Help
Replies
14
Views
797
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
13
Views
451
• Calculus and Beyond Homework Help
Replies
21
Views
3K
• Calculus and Beyond Homework Help
Replies
15
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
2K
• Calculus and Beyond Homework Help
Replies
10
Views
611