# Intersection of two planes (without a given point)

• RosiePosie
PQ + t * (x,y,z)In summary, the two planes intersect in a line PQ and the general equation of a plane through this line can be written in vector form using the cross-product of the two normals. Another method is to eliminate one variable and express the remaining variables in terms of t.
RosiePosie
The Two Planes 2x-7y+5z+1=0 and x+4y-3z=0 intersect in a line PQ. What is the general equation of a plane through this line?

Have done cross product of the two normals to get vector form of the line, and parametric x,y and z. Would like advice on what to do next, Thanks

Any other plane through that line would also have a normal that's perpendicular to the direction vector you found through the cross-product. It will also have to contain a point on the intersection line. Write the general plane as a*x+b*y+c*z+d=0 and translate those conditions into linear equations in a, b, c and d.

Another method would to be to eliminate one variable. Then just that variable be t, then find the other two variables in terms of t.

then r=(x,y,z)

## 1. What is the intersection of two planes?

The intersection of two planes is the line where the two planes meet. It is the set of all points that are common to both planes.

## 2. How do you find the intersection of two planes?

To find the intersection of two planes, you need to solve the system of equations formed by the two planes. This can be done by using elimination or substitution methods.

## 3. Can two planes intersect at more than one point?

Yes, two planes can intersect at more than one point. If the two planes are not parallel, they will intersect at a line. If the two planes are parallel, they will either intersect at infinitely many points or have no intersection at all.

## 4. What does it mean if two planes have no intersection?

If two planes have no intersection, it means that they are parallel and never intersect. This can also mean that the two planes are the same plane.

## 5. How can the intersection of two planes be used in real life?

The intersection of two planes has many real-life applications, such as in architecture and engineering for constructing 3D shapes and structures. It is also used in navigation and aviation for determining the position of an aircraft or ship. Additionally, it is used in computer graphics for creating 3D images and animations.

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