Lines and planes meeting, angles between planes, and distances

Where does the line r = A + t d meet the plane ax + by + cz = k? A point of the line is (x, y, z) = A + t d. Put it into the equation of the plane: the result is a linear equation in t, which is (n · d)t = k − n · A, with n = (a, b, c). Three cases, as in the table. If n · d is not 0, there is one value of t and so exactly one common point: put t back to get the point. If n · d = 0 the direction of the line is perpendicular to the normal, so the line is parallel to the plane; then either the point A is not in the plane (no solution: no common point) or it is (every t works: the line lies in the plane). Next, two planes. Their angle is the angle between their normal vectors, taken acute: cos θ = |n₁ · n₂|/(‖n₁‖‖n₂‖), 0° ≤ θ ≤ 90°. The planes are parallel when the normals are proportional, n₁ = kn₂, and perpendicular when n₁ · n₂ = 0. Different parallel planes do not meet; in other cases two planes meet in a line, studied in the next lesson. Always read normal vectors from the coefficients of x, y, z, and take care with equations written in a different order.