I'm about to teach a lesson on "The angle between planes". I'm having some question on the angle between two planes definition.
Many places defines the angle between two planes as the angle between theirs normal vector.
When the two planes are intersected, we have a property to calculate the angle. That is: draw two lines, one in the first plane, one in the second plane and they all perpendicular to the intersection line. The angle between that 2 lines are equal to the angle of two normal vectors, which also equal the angle of two planes.
To me, the property above feels more intuitive when we talk about the angle between two planes. When people want to calculate the angle between the ladder and the floor, in my opinion, nobody would find the normal vectors and calculate the angle and they often refer to the angle mentioned in the property.
So my question is: What is the intuition of using normal vectors to define the angle between planes? Is there some real life application behind that definition or is it just a mathematical convinience?
And, another little question: How the angle between planes are applied in our real life, especially the case where the two planes are not perpendicular?
Thanks for your reading and thanks for any comments, ideas.