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Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will learn how to work with locations and directions in three-dimensional space, in order to easily describe objects such as lines, planes and curves. This will set the stage for the study of functions of two variables, the graphs of which are surfaces in space. Points in Three-Dimensional Space Previously, we have identiο¬ed a point in the π₯π¦-plane by an ordered pair that consists of two real numbers, an π₯-coordinate and π¦-coordinate, which denote signed distances along the π₯-axis and π¦-axis, respectively, from the origin, which is the point (0, 0). These axes, which are collectively referred to as the coordinate axes, divided the plane into four quadrants. We now generalize these concepts to three-dimensional space, or π₯π¦π§-space. In this space, a point is represented by an ordered triple (π₯, π¦, π§) that consists of three numbers, an π₯-coordiante, a π¦-coordinate, and a π§-coordinate. As in the two-dimensional π₯π¦-plane, these coordinates indicate the signed distance along the coordinate axes, the π₯-axis, π¦-axis and π§-axis, respectively, from the origin, denoted by π, which has coordinates (0, 0, 0). There is a one-to-one correspondence between a point in π₯π¦π§-space and a triple in β3 , which is the set of all ordered triples of real numbers. This correspondence is known as a three-dimensional rectangular coordinate system. Example Figure 1 displays the point (2, 3, 1) in π₯π¦π§-space, denoted by the letter π , along with its projections onto the coordinate planes (described below). The origin is denoted by the letter π. β‘ Planes in Three-Dimensional Space Unlike two-dimensional space, which consists of a single plane, the π₯π¦-plane, three-dimensional space contains inο¬nitely many planes, just as two-dimensional space consists of inο¬nitely many lines. Three planes are of particular importance: the π₯π¦-plane, which contains the π₯- and π¦-axes; the π¦π§-plane, which contains the π¦- and π§-axes; and the π₯π§-plane, which contains the π₯- and π§-axes. Alternatively, the π₯π¦-plane can be described as the set of all points (π₯, π¦, π§) for which π§ = 0. Similarly, the π¦π§-plane is the set of all points of the form (0, π¦, π§), while the π₯π§-plane is the set of all points of the form (π₯, 0, π§). 1 Figure 1: The point (2, 3, 1) in π₯π¦π§-space, denoted by the letter π . The origin is denoted by the letter π. The projections of π onto the coordinate planes are indicated by the diamonds. The dashed lines are line segments perpendicular to the coordinate planes that connect π to its projections. Just as the π₯-axis and π¦-axis divide the π₯π¦-plane into four quadrants, these three planes divide π₯π¦π§-space into eight octants. Within each octant, all π₯-coordiantes have the same sign, as do all π¦-coordinates, and all π§-coordinates. In particular, the ο¬rst octant is the octant in which all three coordinates are positive. Plotting Points in π₯π¦π§-space Graphing in π₯π¦π§-space can be diο¬cult because, unlike graphing in the π₯π¦-plane, depth perception is required. To simplify plotting of points, one can make use of projections onto the coordinate planes. The projection of a point (π₯, π¦, π§) onto the π₯π¦-plane is obtained by connecting the point to the π₯π¦-plane by a line segment that is perpendicular to the plane, and computing the intersection of the line segment with the plane. Later, we will learn more about how to compute projections of points onto planes, but in this 2 relatively simple case, it follows from our working deο¬nition that the projection of the point (π₯, π¦, π§) onto the π₯π¦-plane is the point (π₯, π¦, 0). Similarly, the projection of this point onto the π¦π§-plane is the point (0, π¦, π§), and the projection of this point onto the π₯π§-plane is the point (π₯, 0, π§). Figure 1 illustrates these projections, and how they can be used to plot a point in π₯π¦π§-space. One can ο¬rst plot the point’s projections, which is similar to the task of plotting points in the π₯π¦-plane, and then use line segments originating from these projections and perpendicular to the coordinate planes to “locate” the point in π₯π¦π§-space. The Distance Formula The distance between two points π1 = (π₯1 , π¦1 ) and π2 = (π₯2 , π¦2 ) in the π₯π¦-plane is given by the distance formula, √ π(π1 , π2 ) = (π₯2 − π₯1 )2 + (π¦2 − π¦1 )2 . Similarly, the distance between two points π1 = (π₯1 , π¦1 , π§1 ) and π2 = (π₯2 , π¦2 , π§2 ) in π₯π¦π§-space is given by the following generalization of the distance formula, √ π(π1 , π2 ) = (π₯2 − π₯1 )2 + (π¦2 − π¦1 )2 + (π§2 − π§1 )2 . This can be proved by repeated application of the Pythagorean Theorem. Example The distance between π1 = (2, 3, 1) and π2 = (8, −5, 0) is √ √ √ π(π1 , π2 ) = (8 − 2)2 + (−5 − 3)2 + (0 − 1)2 = 36 + 64 + 1 = 101 ≈ 10.05. β‘ Equations of Surfaces In two dimensions, the solution set of a single equation involving the coordinates π₯ and/or π¦ is a curve. In three dimensions, the solution set of an equation involving π₯, π¦ and/or π§ is a surface. Example The equation π§ = 3 describes a plane that is parallel to the π₯π¦-plane, and is 3 units “above” it; that is, it lies 3 units along the positive π§-axis from the π₯π¦-plane. On the other hand, the equation π₯ = π¦ describes a plane consisting of all points whose π₯- and π¦-coordinates are equal. It is not parallel to any coordinate plane, but it contains the π§-axis, which consists of all points whose π₯- and π¦-coordinates are both zero, and it intersects the π₯π¦-plane at the line π¦ = π₯. β‘ The equation of a sphere with center πΆ = (β, π, β) and radius π is (π₯ − β)2 + (π¦ − π)2 + (π§ − β)2 = π2 . The unit sphere has center π = (0, 0, 0) and radius 1: π₯2 + π¦ 2 + π§ 2 = 1. 3 We now illustrate how to work with equations of spheres. Example The equation of a sphere with center πΆ = (−3, −1, 1) and radius π = 10 is (π₯ − (−3))2 + (π¦ − (−1))2 + (π§ − 1)2 = 102 , or (π₯ + 3)2 + (π¦ + 1)2 + (π§ − 1)2 = 100. Expanding, we obtain π₯2 + π¦ 2 + π§ 2 + 6π₯ + 2π¦ − 2π§ = 89, which obscures the center and radius, but it is still possible to detect that the equation represents a sphere, due to the fact that the π₯2 , π¦ 2 and π§ 2 terms have equal coeο¬cients. β‘ Example The equation 4π₯2 + 4π¦ 2 + 4π§ 2 − 8π₯ − 16π¦ − 16 = 0 describes a sphere, as can be seen by the equal coeο¬cients in front of the π₯2 , π¦ 2 and π§ 2 . To determine the radius and center of the sphere, we complete the square in π₯ and π¦: 0 = 4π₯2 + 4π¦ 2 + 4π§ 2 − 8π₯ − 16π¦ − 16 = 4(π₯2 − 2π₯) + 4(π¦ 2 − 4π¦) + 4π§ 2 − 16 = 4(π₯2 − 2π₯ + 1 − 1) + 4(π¦ 2 − 4π¦ + 4 − 4) + 4π§ 2 − 16 = 4[(π₯ − 1)2 − 1] + 4[(π¦ − 2)2 − 4] + 4π§ 2 − 16 = 4(π₯ − 1)2 + 4(π¦ − 2)2 + 4π§ 2 − 36. Rearranging, we obtain the standard form of the equation of the sphere: (π₯ − 1)2 + (π¦ − 2)2 + π§ 2 = 9, which reveals that the center is at the point πΆ = (1, 2, 0), and the radius is π = 3. β‘ 4 Summary β The three-dimensional rectangular coordinate system is the one-to-one correspondence between each point π in three-dimensional space, or π₯π¦π§-space, and an ordered triple (π₯, π¦, π§) in β3 . The numbers π₯, π¦ and π§ are the π₯-, π¦- and π§-coordinates of π . The origin π is the point with coordinates (0, 0, 0). β The coordinate planes are: the π₯π¦-plane, the set of all points whose π§-coordinate is zero; the π¦π§-plane, the set of all points whose π₯-coordinate is zero; and the π₯π§-plane, the set of all points whose π¦-coordinate is zero. β The projection of a point π = (π₯, π¦, π§) onto the π₯π¦-plane is the point (π₯, π¦, 0). The projection of π onto the π¦π§-plane is the point (0, π¦, π§). The projection of π onto the π₯π§-plane is the point (π₯, 0, π§). β The distance formula states that the distance between two points in π₯π¦π§-space is the square root of the sum of the squares of the diο¬erences between corresponding coordinates. That is, given π1 √ = (π₯1 , π¦1 , π§1 ) and π2 = (π₯2 , π¦2 , π§2 ), the distance between π1 and π2 is given by π(π1 , π2 ) = (π₯2 − π₯1 )2 + (π¦2 − π¦1 )2 + (π§2 − π§1 )2 . β The equation of a sphere with center πΆ = (π₯0 , π¦0 , π§0 ) and radius π is (π₯ − π₯0 )2 + (π¦ − π¦0 )2 + (π§ − π§0 )2 = π2 . β An equation in which π₯2 , π¦ 2 and π§ 2 have the same coeο¬cients describes a sphere; the center and radius can be determined by completing the square in π₯, π¦ and π§. 5