Chapter 1 Topics in Analytic Geometry

•MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 533.6 Plane in 3-SpacePlane Parallel to the Coordinate PlanesThe graph of the equation x = a in an xyz-coordinate system consists of all points of theform (a,y,z), where y and z are arbitrary. One such point is (a,0,0), and all others arein the plane that passes through this point and is parallel to the yz-plane. Similarly, thegraph of y = b is the plane through (0,b,0) that is parallel to the xz-plane, and the graphof z = c is the plane through (0,0,c) that is parallel to the xy-plane.(a,0,0)x = ay = b•(0,b,0)•(0,0,c)z = cPlanes Determined by a Point and a Normal VectorA plane in 3-space can be determined uniquely by specifying a point in the plane and avector perpendicular to the plane. A vector perpendicular to the plane is called a normalto the plane.Suppose that we want to find an equation of the plane passing through P 0 (x 0 ,y 0 ,z 0 )and perpendicular to the vector n = 〈a,b,c〉. Define the vectors r 0 and r asr 0 = 〈x 0 ,y 0 ,z 0 〉 and r = 〈x,y,z〉nP 0 (x 0 ,y 0 ,z 0 ) •r−r 0 • P(x,y,z)r r 0OIt should be evident from the above Figure that the plane consists precisely of those pointsP(x,y,z) for which the vector r−r 0 is orthogonal to n; or, expressed as an equation,n·(r−r 0 ) = 0 (3.29)