Intersection of Planes

When two planes intersects, an intersection line is formed as shown below.

Finding Line of Intersection

To find the line of intersection, recall that the line equation is r(t) = r₀ + tv. The problem is, we do not know what is the vector v. Somehow, if you look into the graph below, we find that v is the vector that is parallel to the cross product of the normal vectors of planes, n₁ and n₂. Therefore, we can just find the equation taking v = n₁ x n₂. The only difficulty is to find a point on the line (P) algebraically. We need the coordinate P to be put into r₀. To do so, we need to solve the simultaneos equations of the two planes.

You may move any un-labelled points to move the planes, and observe the changes on the equations of planes and intersection line. Take note that when you move point P on the line, r₀ changes according to the coordinate of P.

Betty, Created with GeoGebra and Thomas Calculus, Pearson.