Lines and planes meeting, angles between planes, and distances

Two planes that are not parallel meet in a line. To find it, use two facts. The direction: the line lies in both planes, so it is perpendicular to both normals, and d = n₁ × n₂. A point: choose a convenient coordinate, say z = 0 (or any one that makes the equations easy), and solve the two equations for the other two unknowns. The line is r = P + t d. Check that P satisfies both equations and that n₁ · d = n₂ · d = 0. Now the distance from a point A(x₁, y₁, z₁) to a plane. Take any point B in the plane with normal n. The distance is the size of the projection of the vector BA on n, which is |BA · n|/‖n‖. In Cartesian form for the plane ax + by + cz + d = 0 this becomes |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²). The numerator is the equation of the plane with the coordinates of the point put in; the denominator is the size of the normal. In vector form n · r = k, the same distance is |n · A − k|/‖n‖. A zero value means the point is on the plane.