Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 54Intersecting PlanesTwo distinct intersecting plane determine two positive angles of intersection—an (acute)angle θ that satisfies the condition 0 ≤ θ ≤ π/2 and the supplement of that angle.n 1θn 2θθ180 ◦ −θIf n 1 and n 2 are normals to the planes, then depending on the directions of n 1 and n 2 ,the angle θ is either the angle between n 1 and n 2 or the angle between n 1 and −n 2 . In bothcase, we have the following formula for the acute angle θ between the planes:cosθ = |n 1 ·n 2 |‖n 1 ‖‖n 2 ‖(3.33)Example 3.39 Find the acute angle of intersection between the two planesSolution .........2x−4y +4z = 6 and 6x+2y −3z = 4Example 3.40 Find an equation for the line L of intersection of the planes in Example3.39.Solution .........Distance Problems Involving PlanesNext we will consider three basic distance problems in 3-space:• Find the distance between a point and a plane.• Find the distance between two parallel planes.• Find the distance between two skew lines.The three problems are related. If we can find the distance between a point and a plane,then we can find the distance between parallel planes by computing the distance betweenone of the planes and an arbitrary point P 0 in the other plane. Moreover, we can findthe distance between two skew lines by computing the distance between parallel planescontaining them.