Chapter 1 Topics in Analytic Geometry

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Chapter 3Three-Dimensional Space; Vectors3.1 RectangularCoordinatesin3-Space; Spheres; CylindricalSurfacesRectangular Coordinate SystemsTo begin, consider three mutually perpendicular coordinate lines, called the x-axis, they-axis, and the z-axis, positioned so that their origin coincide.zOyxThe three coordinate axes form a three-dimensional rectangular coordinate system (orCartesian coordinate system) The point of intersection of the coordinate axes is calledthe origin of the coordinate system.The coordinate axes, taken in pairs, determine three coordinate planes: the xyplane,the xz-plane, and the yz-plane, which divide space into eight octants. To eachpoint P in 3-space corresponds to ordered triple of real numbers (a,b,c) which measureits directed distances from the three planes. We call a, b, and c the x-coordinate, y-coordinate, and z-coordinate of P, respectively, and we denote the point P by (a,b,c)or by P(a,b,c).The following facts about three-dimensional rectangular coordinate systems:30
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 31Region Descriptionxy-plane Consists of all points of the form (x,y,0)xz-plane Consists of all points of the form (x,0,z)yz-plane Consists of all points of the form (0,y,z)x-axis Consists of all points of the form (x,0,0)y-axis Consists of all points of the form (0,y,0)z-axis Consists of all points of the form (0,0,z)Distance in 3-Space; SpheresRecall that in 2-space the distance d between the points P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ) isd = √ (x 2 −x 1 ) 2 +(y 2 −y 1 ) 2The distance formula in 3-space has the same form, but it has a third term to account forthe added dimension. The distance between the points P 1 (x 1 ,y 1 ,z 1 ) and P 2 (x 2 ,y 2 ,z 2 ) isd = √ (x 2 −x 1 ) 2 +(y 2 −y 1 ) 2 +(z 2 −z 1 ) 2Example 3.1 Find the distance d between the points (2,−3,4) and (−3,2,−5).Solution .........Recall that the standard equation of a circle in 2-space that has center (x 0 ,y 0 ) andradius r is(x−x 0 ) 2 +(y −y 0 ) 2 = r 2Analogously, standard equation of the sphere in 3-space that has center (x 0 ,y 0 ,z 0 )and radius r is(x−x 0 ) 2 +(y −y 0 ) 2 +(z −z 0 ) 2 = r 2 (3.1)Ifthetermsin(3.1)areexpandedandliketermsarecollected, thentheresultingequationhas the formx 2 +y 2 +z 2 +Gx+Hy +Iz +J = 0 (3.2)Example 3.2 Find the center and radius of the sphereSolution .........x 2 +y 2 +z 2 −10x−8y −12z +68 = 0In general, completing the squares in (3.2) produces an equation of the form(x−x 0 ) 2 +(y −y 0 ) 2 +(z −z 0 ) 2 = k 2• If k > 0, then the graph of this equation is a sphere with center (x 0 ,y 0 ,z 0 ) and radius√k.• If k = 0, then the sphere has radius zero, so the graph is the single point (x 0 ,y 0 ,z 0 ).
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Chapter 1 Topics in Analytic Geometry

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