Line of intersection of two planes

[VenaCava]

Hi,
I am having difficutly figuring out why the cross product of the normal vectors of each plane gives the direction vector of the line of intersection. Anyone care to try to explain?


Thanks!

 

[mathman]

The line of intersection lies in both planes. The normal to a plane is (by definition) perpendicular to any line in the plane. The cross product then gives you a line perpendicular to both normals, so that it must be parallel to the line of intersection.

 

[HallsofIvy]

The line of intersection lies in both planes. The normal to a plane is (by definition) perpendicular to any line in the plane. The cross product then gives you a line perpendicular to both normals,

therefore lieing in both planes, therefore along the line of intersection

so that it must be parallel to the line of intersection.

 

[mathman]

therefore lieing in both planes, therefore along the line of intersection

The cross product is a vector, NOT a line is space - that is, it has a direction but no position. Therefore it doesn't lie anywhere, but it is parallel to the line of intersection.

 

[CellCoree]

I can help explain the concept of the line of intersection of two planes.

First, it's important to understand that planes are two-dimensional surfaces that extend infinitely in all directions. When two planes intersect, they create a line where the two surfaces meet. This line is known as the line of intersection.

Now, let's consider the normal vectors of each plane. A normal vector is a vector that is perpendicular to the plane. This means that it is at a 90-degree angle to the surface of the plane. When two planes intersect, their normal vectors will also intersect at a 90-degree angle, creating a cross shape.

The cross product of two vectors is a mathematical operation that results in a vector that is perpendicular to both vectors. In this case, the cross product of the two normal vectors will give us a vector that is perpendicular to both planes. This vector will be the direction vector of the line of intersection.

In simpler terms, the cross product of the normal vectors gives us a vector that is perpendicular to both planes, and therefore, perpendicular to the line of intersection. This vector represents the direction in which the line of intersection is pointing.

I hope this explanation helps clarify the concept of the line of intersection and how the cross product of the normal vectors relates to it. If you have any further questions, please don't hesitate to ask.