1.6.3—Lines and planes

A line is r=a+λd; a plane is n·(r−b)=0. A line is parallel to a plane when n·d=0, lies in it when a point also satisfies the plane equation, and intersects it otherwise.

To find a line–plane intersection, substitute the line into the plane equation to solve λ. For two planes, the cross product of their normals gives the direction of their line of intersection when the normals are not parallel.

For r=(1,0,0)+λ(1,2,−1) and plane x+y+z=3, substitution gives 1+λ+2λ−λ=3, so λ=2/2=1 and the intersection point is (2,2,−1).

Parallel normals do not prove two planes are the same; compare a point or the constants, and keep line direction distinct from position.