The Angle Between Planes - Concept
We can use knowledge of how to find the angle between vectors to help us find the angle between planes. We can use the dot product to find the angle between planes, or determine whether or not the planes are perpendicular or parallel. In order to find the angle between planes we must understand finding vectors normal to planes.
Let's talk about the angle between planes. Now if you have 2 planes in space there are 2 possibilities either they intersect or they don't ever intersect, if they do intersect, they intersect in a line and they'll make an angle with each other. Now the way to measure the angle between planes and by that I mean this angle is to measure the angle between their normal. You just have to be careful because if this angle is small, then this angle will be small but if this angle is large, then this angle will be large too and the angle up here is actually going to be the smaller of the 2 and we always take the smaller of the 2 angles between theta and 180 minus theta 180 degrees minus theta. So the smaller one of these 2 angles will be the angle between the 2 planes.
Now how do you tell if they are perpendicular, it's really easy you just take the 2 normal vectors and perform their dot product. If the dot product is 0 then the normal vectors are perpendicular and that means the planes are also perpendicular. So what if they don't intersect? Well if 2 planes don't intersect then they have to be parallel or I mean they do intersect and they intersect everywhere they're at the same plane and if they don't intersect at all they're parallel and the test for a parallel planes is that one vector one normal is scalar multiple of the other. So if you recognize that one of these normals is a scalar multiple say 2 times the other or minus 1 times the other then the planes are parallel.