Angle between Planes

So far the angle between two lines has been found by comparing their direction vectors. A plane, unlike a line, does not carry a single direction vector of its own — instead it is characterised by a normal vector, the vector perpendicular to every line lying in the plane. This suggests a natural way to compare the inclination of two planes: compare their normals. This article develops that idea in two related situations. First, in 6.5.1, the angle between two planes is measured by measuring the angle between their normal vectors. Second, in 6.5.2, the angle between a line and a plane is measured by comparing the line's direction vector with the plane's normal vector — but here a small twist appears, because the angle a line makes with a plane is conventionally measured from the plane itself and not from the normal, so the formula that emerges is a sine formula rather than a cosine formula …