8.6 Three-Dimensional Cartesian Coordinate System

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1 SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 69 What ou ll learn about Three-Dimensional Cartesian Coordinates Distance and Midpoint Formulas Equation of a Sphere Planes and Other Surfaces Vectors in Space Lines in Space... and wh This is the analtic geometr of our phsical world. 8.6 Three-Dimensional Cartesian Coordinate Sstem Three-Dimensional Cartesian Coordinates In Sections P. and P.4, we studied Cartesian coordinates and the associated basic formulas and equations for the two-dimensional plane; we now etend these ideas to three-dimensional space. In the plane, we used two aes and ordered pairs to name points; in space, we use three mutuall perpendicular aes and ordered triples of real numbers to name points (Figure 8.46). constant (0, 0, ) (0,, ) (, 0, ) (, 0, 0) constant P(,, ) (0,, 0) constant (,, 0) FIGURE 8.46 The point P1,, in Cartesian space. Notice that Figure 8.46 ehibits several important features of the three-dimensional Cartesian coordinate sstem: 0 0 FIGURE 8.47 The coordinate planes divide space into eight octants. (0, 0, 0) Origin 0 The aes are labeled,, and, and these three coordinate aes form a righthanded coordinate frame: When ou hold our right hand with fingers curving from the positive -ais toward the positive -ais, our thumb points in the direction of the positive -ais. A point P in space uniquel corresponds to an ordered triple 1,, of real numbers. The numbers,, and are the Cartesian coordinates of P. Points on the aes have the form 1, 0, 0, 10,, 0, or 10, 0,, with 1, 0, 0 on the -ais, 10,, 0 on the -ais, and 10, 0, on the -ais. In Figure 8.47, the aes are paired to determine the coordinate planes: The coordinate planes are the -plane, the -plane, and the -plane, and have equations = 0, = 0, and = 0, respectivel. Points on the coordinate planes have the form 1,, 0, 1, 0,, or 10,,, with 1,, 0 on the -plane, 1, 0, on the -plane, and 10,, on the -plane. The coordinate planes meet at the origin, (0, 0, 0). The coordinate planes divide space into eight regions called octants, with the first octant containing all points in space with three positive coordinates.

2 630 CHAPTER 8 Analtic Geometr in Two and Three Dimensions EXAMPLE 1 Locating a Point in Cartesian Space Draw a sketch that shows the point 1, 3, 5. SOLUTION To locate the point 1, 3, 5, we first sketch a right-handed three-dimensional coordinate frame. We then draw the planes =, = 3, and = 5, which parallel the coordinate planes = 0, = 0, and = 0, respectivel. The point 1, 3, 5 lies at the intersection of the planes =, = 3, and = 5, as shown in Figure Now tr Eercise 1. (0, 0, 5) (, 3, 5) Line 3, 5 Plane 5 Line, 5 Plane Plane 3 (, 0, 0) 0 (0, 3, 0) Line, 3 FIGURE 8.48 The planes =, = 3, and = 5 determine the point 1, 3, 5. (Eample 1) Distance and Midpoint Formulas The distance and midpoint formulas for space are natural etensions of the corresponding formulas for the plane. Distance Formula (Cartesian Space) The distance d1p, Q between the points P1 1, 1, 1 and Q1,, in space is d1p, Q = Just as in the plane, the coordinates of the midpoint of a line segment are the averages for the coordinates of the endpoints of the segment. Midpoint Formula (Cartesian Space) The midpoint M of the line segment PQ with endpoints P1 1, 1, 1 and Q1,, is M = a 1 +, 1 +, 1 + b.

3 SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 631 EXAMPLE Calculating a Distance and Finding a Midpoint Find the distance between the points P1-, 3, 1 and Q14, -1, 5, and find the midpoint of line segment PQ. SOLUTION The distance is given b The midpoint is d1p, Q = = = 168 L 8.5 M = a - + 4, 3-1, b = 11, 1, 3. Now tr Eercises 5 and 9. Drawing Lesson How to Draw Three-Dimensional Objects to Look Three-Dimensional 1. Make the angle between the positive -ais and the positive -ais large enough. This Not this. Break lines. When one line passes behind another, break it to show that it doesn t touch and that part of it is hidden. A D A D A D C C C B B B Intersecting CD behind AB AB behind CD 3. Dash or omit hidden portions of lines. Don t let the line touch the boundar of the parallelogram that represents the plane, unless the line lies in the plane. Line below plane Line above plane Line in plane 4. Spheres: Draw the sphere first (outline and equator); draw aes, if an, later. Use line breaks and dashed lines. Hidden part dashed Break A contact dot sometimes helps Break Sphere first Aes later

4 63 CHAPTER 8 Analtic Geometr in Two and Three Dimensions Equation of a Sphere A sphere is the three-dimensional analogue of a circle: In space, the set of points that lie a fied distance from a fied point is a sphere. The fied distance is the radius, and the fied point is the center of the sphere. The point P1,, is a point of the sphere with center (h, k, l) and radius r if and onl if 1 - h k l = r. Squaring both sides gives the standard equation shown below. Standard Equation of a Sphere A point P1,, is on the sphere with center 1h, k, l and radius r if and onl if 1 - h k + ( - l = r. EXAMPLE 3 Finding the Standard Equation of a Sphere The standard equation of the sphere with center 1, 0, -3 and radius 7 is = 49. Now tr Eercise 13. Planes and Other Surfaces In Section P.4, we learned that ever line in the Cartesian plane can be written as a firstdegree (linear) equation in two variables; that is, ever line can be written as A + B + C = 0, where A and B are not both ero. Conversel, ever first-degree equation in two variables represents a line in the Cartesian plane. In an analogous wa, ever plane in Cartesian space can be written as a first-degree equation in three variables: Equation for a Plane in Cartesian Space Ever plane can be written as A + B + C + D = 0, where A, B, and C are not all ero. Conversel, ever first-degree equation in three variables represents a plane in Cartesian space. (0, 0, 3) (0, 4, 0) (5, 0, 0) = 60 FIGURE 8.49 The intercepts 15, 0, 0, 10, 4, 0, and 10, 0, 3 determine the plane = 60. (Eample 4) EXAMPLE 4 Sketching a Plane in Space Sketch the graph of = 60. SOLUTION Because this is a first-degree equation, its graph is a plane. Three points determine a plane. To find three points, we first divide both sides of = 60 b 60: = 1 In this form, it is eas to see that the points 15, 0, 0, 10, 4, 0, and 10, 0, 3 satisf the equation. These are the points where the graph crosses the coordinate aes. Figure 8.49 shows the completed sketch. Now tr Eercise 17.

5 SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 633 Equations in the three variables,, and generall graph as surfaces in three-dimensional space. Just as in the plane, second-degree equations are of particular interest. Recall that second-degree equations in two variables ield conic sections in the Cartesian plane. In space, second-degree equations in three variables ield quadric surfaces: The paraboloids, ellipsoids, and hperboloids of revolution that have special reflective properties are all quadric surfaces, as are such eotic-sounding surfaces as hperbolic paraboloids and elliptic hperboloids. Other surfaces of interest include graphs of functions of two variables, whose equations have the form = ƒ1,. Some eamples are = ln, = sin 1, and = The last equation graphs as a hemisphere (see Eercise 63). Equations of the form = ƒ1, can be graphed using some graphing calculators and most computer algebra software. Quadric surfaces and functions of two variables are studied in most universit-level calculus course sequences. v 3 v 1, v, v 3 v 0, 0, 1 k 0, 1, 0 v i j 1, 0, 0 v 1 FIGURE 8.50 The vector v = 8v 1, v, v 3 9. Vectors in Space In space, just as in the plane, the sets of equivalent directed line segments (or arrows) are vectors. The are used to represent forces, displacements, and velocities in three dimensions. In space, we use ordered triples to denote vectors: v = 8v 1, v, v 3 9 The ero vector is 0 = 80, 0, 09, and the standard unit vectors are i = 81, 0, 09, j = 80, 1, 09, and k = 80, 0, 19. As shown in Figure 8.50, the vector v can be epressed in terms of these standard unit vectors: v = 8v 1, v, v 3 9 = v 1 i + v j + v 3 k The vector v that is represented b the arrow from P1a, b, c to Q1,, is v = PQ! = 8 - a, - b, - c9 = 1 - ai bj ck. A vector v = 8v 1, v, v 3 9 can be multiplied b a scalar (real number) c as follows: cv = c8v 1, v, v 3 9 = 8cv 1, cv, cv 3 9 Man other properties of vectors etend in a natural wa when we move from two to three dimensions: Vector Relationships in Space For vectors v = 8v 1, v, v 3 9 and w = 8w 1, w, w 3 9, Equalit: v = w if and onl if v 1 = w 1, v = w, and v 3 = w 3 Addition: v + w = 8v 1 + w 1, v + w, v 3 + w 3 9 Subtraction: v - w = 8v 1 - w 1, v - w, v 3 - w 3 9 Magnitude: ƒvƒ = v 1 + v + v 3 Dot product: v # w = v1 w 1 + v w + v 3 w 3 Unit vector: u = v/ƒvƒ, v Z 0, is the unit vector in the direction of v. EXAMPLE 5 Computing with Vectors (a) (b) (c) (d) (e) 38-, 1, 49 = 83 # -, 3 # 1, 3 # 49 = 8-6, 3, 19 80, 6, , 5, 89 = 80-5, 6 + 5, = 8-5, 11, 19 81, -3, 49-8-, -4, 59 = 81 +, , 4-59 = 83, 1, -19 ƒ 8, 0, - 69 ƒ = = 140 L , 3, -19 # 8-6,, 39 = 5 # # # 3 = = -7 Now tr Eercises 3 6.

6 634 CHAPTER 8 Analtic Geometr in Two and Three Dimensions EXAMPLE 6 Using Vectors in Space A jet airplane just after takeoff is pointed due east. Its air velocit vector makes an angle of 30 with flat ground with an airspeed of 50 mph. If the wind is out of the southeast at 3 mph, calculate a vector that represents the plane s velocit relative to the point of takeoff. SOLUTION Let i point east, j point north, and k point up. The plane s air velocit is a = 850 cos 30, 0, 50 sin 30 9 L , 0, 159, and the wind velocit, which is pointing northwest, is w = 83 cos 135, 3 sin 135, 09 L 8-.67,.67, 09. The velocit relative to the ground is v = a + w, so v L , 0, ,.67, 09 L ,.63, 159 = i +.63j + 15k Now tr Eercise 33. v = a, b, c P(,, ) P 0 ( 0, 0, 0 ) FIGURE 8.51 The line / is parallel to the direction vector v = 8a, b, c9. In Eercise 64, ou will be asked to interpret the meaning of the velocit vector obtained in Eample 6. Lines in Space We have seen that first-degree equations in three variables graph as planes in space. So how do we get lines? There are several was. First notice that to specif the -ais, which is a line, we could use the pair of first-degree equations = 0 and = 0. As alternatives to using a pair of Cartesian equations, we can specif an line in space using one vector equation, or a set of three parametric equations. Suppose / is a line through the point P 0 1 0, 0, 0 and in the direction of a nonero vector v = 8a, b, c9 (Figure 8.51). Then for an point on, P 0 P! P1,, 9 / = tv for some real number! t. The vector v is a direction vector for line /. If r = OP! = 8,, 9 and r 0 = OP 0 = 8 0, 0, 0 9, then r - r 0 = tv. So an equation of the line / is r = r 0 + tv. Equations for a Line in Space If / is a line through the point P 0 1 0, 0, 0 in the direction of a nonero vector v = 8a, b, c9, then a point P1,, is on / if and onl if Vector form: r = r 0 + tv, where r = 8,, 9 and r 0 = 8 0, 0, 0 9; or Parametric form: = 0 + at, = 0 + bt, and = 0 + ct, where t is a real number. EXAMPLE 7 Finding Equations for a Line The line through P 0 14, 3, -1 with direction vector v = 8-,, 79 can be written in vector form as in parametric form as r = 84, 3, t8-,, 79; or = 4 - t, = 3 + t, and = t. Now tr Eercise 35.

7 SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 635 EXAMPLE 8 Finding Equations for a Line Using the standard unit vectors i, j, and k, write a vector equation for the line containing the points A13, 0, - and B1-1,, -5, and compare it to the parametric equations for the line. SOLUTION The line is in the direction of v Using r 0 = OA! = AB! = , - 0, = 8-4,, -39., the vector equation of the line becomes: r = r 0 + tv 8,, 9 = 83, 0, -9 + t8-4,, -39 8,, 9 = 83-4t, t, - - 3t9 i + j + k = 13-4ti + tj tk The parametric equations are the three component equations = 3-4t, = t, and = - - 3t. Now tr Eercise 41. QUICK REVIEW 8.6 (For help, go to Sections 6.1 and 6.3.) Eercise numbers with a gra background indicate problems that the authors have designed to be solved without a calculator. In Eercises 1 3, let P1, and Q1, -3 be points in the -plane. 1. Compute the distance between P and Q.. Find the midpoint of the line segment PQ. 3. If P is 5 units from Q, describe the position of P. In Eercises 4 6, let v = 8-4, 59 = -4i + 5j be a vector in the -plane. 4. Find the magnitude of v. 5. Find a unit vector in the direction of v. 6. Find a vector 7 units long in the direction of -v. 7. Give a geometric description of the graph of = 5 in the -plane. 8. Give a geometric description of the graph of = - t, = -4 + t in the -plane. 9. Find the center and radius of the circle = 0 in the -plane. 10. Find a vector from P1, 5 to Q1-1, -4 in the -plane. SECTION 8.6 EXERCISES In Eercises 1 4, draw a sketch that shows the point , 4,. 1, -3, , -, , 3, -5 In Eercises 5 8, compute the distance between the points ,, 5, 13, -4, , -1, -8, 16, -3, a, b, c, 11, -3, 8. 1,,, 1p, q, r In Eercises 9 1, find the midpoint of the segment PQ. 9. P1-1,, 5, Q13, -4, P1, -1, -8, Q16, -3, P1,,, Q1-, 8, 6 1. P1-a, -b, -c, Q13a, 3b, 3c In Eercises 13 16, write an equation for the sphere with the given point as its center and the given number as its radius , -1, -, , 5, 8, , -3,, 1a, a p, q, r, 6 In Eercises 17, sketch a graph of the equation. Label all intercepts = = = = = 6. = 3 In Eercises 3 3, evaluate the epression using r = 81, 0, -39, v = 8-3, 4, -59, and w = 84, -3, r + v 4. r - w 5. v# w 6. ƒwƒ 7. r # 1v + w 8. 1r# v + 1r # w 9. w/ƒwƒ 30. i# r 31. 8i # v, j # v, k # v9 3. 1r# vw

8 636 CHAPTER 8 Analtic Geometr in Two and Three Dimensions In Eercises 33 and 34, let i point east, j point north, and k point up. 33. Three-Dimensional Velocit An airplane just after takeoff is headed west and is climbing at a 0 angle relative to flat ground with an airspeed of 00 mph. If the wind is out of the northeast at 10 mph, calculate a vector v that represents the plane s velocit relative to the point of takeoff. 34. Three-Dimensional Velocit A rocket soon after takeoff is headed east and is climbing at an 80 angle relative to flat ground with an airspeed of 1,000 mph. If the wind is out of the southwest at 8 mph, calculate a vector v that represents the rocket s velocit relative to the point of takeoff. In Eercises 35 38, write the vector and parametric forms of the line through the point P 0 in the direction of v. 35. P 0 1, -1, 5, v = 83,, P 0 1-3, 8, -1, v = 8-3, 5, P 0 16, -9, 0, v = 81, 0, P 0 10, -1, 4, v = 80, 0, 19 In Eercises 39 48, use the points A1-1,, 4, B10, 6, -3, and C1, -4, Find the distance from A to the midpoint of BC. 40. Find the vector from A to the midpoint of BC. 41. Write a vector equation of the line through A and B. 4. Write a vector equation of the line through A and the midpoint of BC. 43. Write parametric equations for the line through A and C. 44. Write parametric equations for the line through B and C. 45. Write parametric equations for the line through B and the midpoint of AC. 46. Write parametric equations for the line through C and the midpoint of AB. 47. Is ABC equilateral, isosceles, or scalene? 48. If M is the midpoint of BC, what is the midpoint of AM? In Eercises 49 5, (a) sketch the line defined b the pair of equations, and (b) Writing to Learn give a geometric description of the line, including its direction and its position relative to the coordinate frame. 49. = 0, = = 0, = 51. = -3, = 0 5. = 1, = Write a vector equation for the line through the distinct points P1 1, 1, 1 and Q1,,. 54. Write parametric equations for the line through the distinct points P1 1, 1, 1 and Q1,,. 55. Generaliing the Distance Formula Prove that the distance d1p, Q between the points P1 1, 1, 1 and Q1,, in space is b using the point R1,, 1, the two-dimensional distance formula within the plane = 1, the one-dimensional distance formula within the line r = 8,, t9, and the Pthagorean Theorem. [Hint: A sketch ma help ou visualie the situation.] 56. Generaliing a Propert of the Dot Product Prove u # u = ƒuƒ where u is a vector in three-dimensional space. Standardied Test Questions 57. True or False + 4 = 1 represents a surface in space. Justif our answer. 58. True or False The parametric equations = 1 + 0t, = - 0t, = t represent a line in space. Justif our answer. In Eercises 59 6, ou ma use a graphing calculator to solve the problem. 59. Multiple Choice A first-degree equation in three variables graphs as (A) a line. (B) a plane. (C) a sphere. (D) a paraboloid. (E) an ellipsoid. 60. Multiple Choice Which of the following is not a quadric surface? (A) A plane (B) A sphere (C) An ellipsoid (D) An elliptic paraboloid (E) A hperbolic paraboloid 61. Multiple Choice If v and w are vectors and c is a scalar, which of these is a scalar? (A) v + w (B) v - w (C) v # w (D) cv (E) ƒvƒw 6. Multiple Choice The parametric form of the line r = 8, -3, 09 + t81, 0, -19 is (A) = - 3t, = 0 + 1t, = 0-1t. (B) = t, = t, = 0-1t. (C) = 1 + t, = 0-3t, = t. (D) = 1 + t, = -3, = -1t. (E) = + t, = -3, = -t. Eplorations 63. Group Activit Writing to Learn The figure shows a graph of the ellipsoid /9 + /4 + /16 = 1 drawn in a bo using Mathematica computer software. (a) Describe its cross sections in each of the three coordinate planes, that is, for = 0, = 0, and = 0. In our description, include the name of each cross section and its position relative to the coordinate frame.

9 CHAPTER 8 Ke Ideas 637 (b) Eplain algebraicall wh the graph of = Revisiting Eample 6 Read Eample 6. Then using is half of a sphere. What is the equation of the related v = i +.63j + 15k, establish the following: whole sphere? (a) The plane s compass bearing is (c) B hand, sketch the graph of the hemisphere (b) Its speed downrange (that is, ignoring the vertical component) is 195. mph. = Check our sketch using a 3D grapher if ou have access to one. (c) The plane is climbing at an angle of (d) Eplain how the graph of an ellipsoid is related to the (d) The plane s overall speed is 31.8 mph. graph of a sphere and wh a sphere is a degenerate ellipsoid Etending the Ideas The cross product u : v of the vectors u = u 1 i + u j + u 3 k and v = v 1 i + v j + v 3 k is i j k u * v = 3 u 1 u u 3 3 v 1 v v 3 = 1u v 3 - u 3 v i + 1u 3 v 1 - u 1 v 3 j + 1u 1 v - u v 1 k. Use this definition in Eercises , -, 39 * 8-, 1, , -1, 9 * 81, -3, Prove that i * j = k. 68. Assuming the theorem about angles between vectors (Section 6.) holds for three-dimensional vectors, prove that u * v is perpendicular to both u and v if the are nonero. 4 CHAPTER 8 Ke Ideas Properties, Theorems, and Formulas Parabolas with Verte (0, 0) 583 Parabolas with Verte (h, k) 584 Ellipses with Center (0, 0) 59 Ellipses with Center (h, k) 594 Hperbolas with Center (0, 0) 604 Hperbolas with Center (h, k) 605 Translation Formulas 614 Rotation Formulas 614 Coefficients for a Conic in a Rotated Sstem 615 Angle of Rotation to Eliminate the Cross-Product Term 616 Discriminant Test 617 Focus Directri Eccentricit Relationship 60 Polar Equations for Conics 6 Ellipse with Eccentricit e and Semimajor Ais a 65 Distance Formula (Cartesian Space) 630 Midpoint Formula (Cartesian Space) 630 Standard Equation of a Sphere 63 Equation for a Plane in Cartesian Space 63 Vector Relationships in Space 633 Equations for a Line in Space 634 Procedures How to Sketch the Ellipse = a b How to Sketch the Hperbola = a b Translation of Aes Rotation of Aes How to Draw Three-Dimensional Objects to Look Three-Dimensional

Three-Dimensional Coordinates

CHAPTER Three-Dimensional Coordinates Three-dimensional movies superimpose two slightl different images, letting viewers with polaried eeglasses perceive depth (the third dimension) on a two-dimensional