Three-Dimensional Coordinate Systems

1 Jim LambersMAT 169 Fall Semester 2009-10 lecture 17 NotesThese notes correspond to Section in the Coordinate SystemsOver the course of the next several lectures, we will learn how to work with locations and directionsin Three-Dimensional space, in order to easily describe objects such as lines, planes and curves. Thiswill set the stage for the study of functions of two variables, the graphs of which are surfaces in Three-Dimensional SpacePreviously, we have identified a point in the -plane by an ordered pair that consists of two realnumbers, an - Coordinate and - Coordinate , which denote signed distances along the -axis and -axis, respectively, from the origin, which is the point (0,0). These axes, which are collectivelyreferred to as the Coordinate axes, divided the plane into four now generalize these concepts to Three-Dimensional space, or -space. In this space, apoint is represented by anordered triple( , , ) that consists of three numbers, an -coordiante, a - Coordinate , and a - Coordinate .

2 As in the two-dimensional -plane, these coordinates indicatethe signed distance along thecoordinate axes, the -axis, -axis and -axis, respectively, from theorigin, denoted by , which has coordinates (0,0,0). There is a one-to-one correspondence betweena point in -space and a triple in 3, which is the set of all ordered triples of real numbers. Thiscorrespondence is known as athree-dimensional rectangular Coordinate 1 displays the point (2,3,1) in -space, denoted by the letter , along with itsprojections onto the Coordinate planes (described below). The origin is denoted by the letter . Planes in Three-Dimensional SpaceUnlike two-dimensional space, which consists of a single plane, the -plane, three-dimensionalspace contains infinitely many planes, just as two-dimensional space consists of infinitely manylines. Three planes are of particular importance: the -plane, which contains the - and -axes;the -plane, which contains the - and -axes; and the -plane, which contains the - and , the -plane can be described as the set of all points ( , , ) for which = , the -plane is the set of all points of the form (0, , ), while the -plane is the set ofall points of the form ( ,0, ).

3 1 Figure 1: The point (2,3,1) in -space, denoted by the letter . The origin is denoted bythe letter . The projections of onto the Coordinate planes are indicated by the dashed lines are line segments perpendicular to the Coordinate planes that connect to as the -axis and -axis divide the -plane into four quadrants, these three planes divide -space into eightoctants. Within each octant, all -coordiantes have the same sign, as do all - coordinates , and all - coordinates . In particular, thefirst octantis the octant in which all threecoordinates are Points in -spaceGraphing in -space can be difficult because, unlike graphing in the -plane, depth perceptionis required. To simplify plotting of points, one can make use ofprojectionsonto the coordinateplanes. The projection of a point ( , , ) onto the -plane is obtained by connecting the point tothe -plane by a line segment that is perpendicular to the plane, and computing the intersectionof the line segment with the , we will learn more about how to compute projections of points onto planes, but in this2relatively simple case, it follows from our working definition that the projection of the point ( , , )onto the -plane is the point ( , ,0).

4 Similarly, the projection of this point onto the -plane isthe point (0, , ), and the projection of this point onto the -plane is the point ( ,0, ). Figure 1illustrates these projections, and how they can be used to plot a point in -space. One can firstplot the point s projections, which is similar to the task of plotting points in the -plane, and thenuse line segments originating from these projections and perpendicular to the Coordinate planes to locate the point in Distance FormulaThe distance between two points 1= ( 1, 1) and 2= ( 2, 2) in the -plane is given by thedistance formula, ( 1, 2) =p( 2 1)2+ ( 2 1) , the distance between two points 1= ( 1, 1, 1) and 2= ( 2, 2, 2) in -space isgiven by the following generalization of the distance formula, ( 1, 2) =p( 2 1)2+ ( 2 1)2+ ( 2 1) can be proved by repeated application of the Pythagorean distance between 1= (2,3,1) and 2= (8, 5,0) is ( 1, 2) =p(8 2)2+ ( 5 3)2+ (0 1)2= 36 + 64 + 1 = 101 Equations of SurfacesIn two dimensions, the solution set of a single equation involving the coordinates and/or is acurve.

5 In three dimensions, the solution set of an equation involving , and/or is a equation = 3 describes a plane that is parallel to the -plane, and is 3 units above it; that is, it lies 3 units along the positive -axis from the -plane. On the other hand,the equation = describes a plane consisting of all points whose - and - coordinates are is not parallel to any Coordinate plane, but it contains the -axis, which consists of all pointswhose - and - coordinates are both zero, and it intersects the -plane at the line = . The equation of a sphere with center = ( , , ) and radius is( )2+ ( )2+ ( )2= spherehas center = (0,0,0) and radius 1: 2+ 2+ 2= now illustrate how to work with equations of equation of a sphere with center = ( 3, 1,1) and radius = 10 is( ( 3))2+ ( ( 1))2+ ( 1)2= 102,or( + 3)2+ ( + 1)2+ ( 1)2= , we obtain 2+ 2+ 2+ 6 + 2 2 = 89,which obscures the center and radius, but it is still possible to detect that the equation representsa sphere, due to the fact that the 2, 2and 2terms have equal coefficients.

6 ExampleThe equation4 2+ 4 2+ 4 2 8 16 16 = 0describes a sphere, as can be seen by the equal coefficients in front of the 2, 2and 2. Todetermine the radius and center of the sphere, we complete the square in and :0 = 4 2+ 4 2+ 4 2 8 16 16= 4( 2 2 ) + 4( 2 4 ) + 4 2 16= 4( 2 2 + 1 1) + 4( 2 4 + 4 4) + 4 2 16= 4[( 1)2 1] + 4[( 2)2 4] + 4 2 16= 4( 1)2+ 4( 2)2+ 4 2 , we obtain the standard form of the equation of the sphere:( 1)2+ ( 2)2+ 2= 9,which reveals that the center is at the point = (1,2,0), and the radius is = 3. 4 Summary The Three-Dimensional rectangular Coordinate system is the one-to-one correspondence be-tween each point in Three-Dimensional space, or -space, and an ordered triple ( , , )in 3. The numbers , and are the -, - and - coordinates of . The origin is thepoint with coordinates (0,0,0). The Coordinate planes are: the -plane, the set of all points whose - Coordinate is zero; the -plane, the set of all points whose - Coordinate is zero; and the -plane, the set of allpoints whose - Coordinate is zero.

7 The projection of a point = ( , , ) onto the -plane is the point ( , ,0). The projectionof onto the -plane is the point (0, , ). The projection of onto the -plane is thepoint ( ,0, ). The distance formula states that the distance between two points in -space is the squareroot of the sum of the squares of the differences between corresponding coordinates . Thatis, given 1= ( 1, 1, 1) and 2= ( 2, 2, 2), the distance between 1and 2is given by ( 1, 2) =p( 2 1)2+ ( 2 1)2+ ( 2 1)2. The equation of a sphere with center = ( 0, 0, 0) and radius is ( 0)2+ ( 0)2+( 0)2= 2. An equation in which 2, 2and 2have the same coefficients describes a sphere; the centerand radius can be determined by completing the square in , and .5