Plane Intersections and Angles (HL only)

For a line r= a+t d and plane r. n=k, substitute the line into the plane: a. n+t(d. n)=k. **Condition** | **Line–plane relationship** d. n 0 | One intersection point d. n=0, a. n=k | Line contained in the plane d. n=0, a. n!= k | Line parallel to the plane without meeting it The line–plane angle alpha satisfies sinalpha= | d. n|| d|| n|. The smaller angle theta between two planes satisfies costheta= | n_1. n_2|| n_1|| n_2|. Both lie in [0, 2]. Line–plane angle Section through the line and its perpendicular projection. Plane–plane angle Perpendicular cross-section through the intersection line. Two nonparallel planes meet in a line with direction n_1 x n_2. Find one common point by solving the equations, then write the line through that point. Parallel normals require a check for coincident versus distinct parallel planes. Three planes Solve all three Cartesian equations together. One unique solution represents one common point; one free parameter represents a common line; two free parameters represent the same plane; inconsistency means there is no common point. Pairwise intersections do not guarantee a point common to all three. Example The planes x+y+z=3, x-y=1, z=0 meet at (2,1,0). Replacing the third equation by 2x+2y+2z=6 gives a common line; replacing it by 2x+2y+2z=7 gives no common point.