Why does the cross product of normal vectors give the line of intersection direction?

[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.