Analytic Geometry in Three Dimensions

Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used in chemistr to help understand the structure of crstals. For eample, isometric crstals are shaped like cubes Arnold Fisher/Photo Researchers, Inc. SELECTED APPLICATIONS Three-dimensional analtic geometr concepts have man real-life applications. The applications listed below represent a small sample of the applications in this chapter. Crstals, Eercise 6, page 88 Geograph, Eercise 66, page 89 Tension, Eercise, page 86 Torque, Eercise, page 8 Data Analsis: Milk Consumption, Eercise 7, page 8 Mechanical Design, Eercise 8, page 8 Moment of a Force, Eercise, page 89 8

8 Chapter Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem What ou should learn Plot points in the threedimensional coordinate sstem. Find distances between points in space and find midpoints of line segments joining points in space. Write equations of spheres in standard form and find traces of surfaces in space. Wh ou should learn it The three-dimensional coordinate sstem can be used to graph equations that model surfaces in space, such as the spherical shape of Earth, as shown in Eercise 66 on page 89. The Three-Dimensional Coordinate Sstem Recall that the Cartesian plane is determined b two perpendicular number lines called the -ais and the -ais. These aes, together with their point of intersection (the origin), allow ou to develop a two-dimensional coordinate sstem for identifing points in a plane. To identif a point in space, ou must introduce a third dimension to the model. The geometr of this three-dimensional model is called solid analtic geometr. You can construct a three-dimensional coordinate sstem b passing a -ais perpendicular to both the - and -aes at the origin. Figure. shows the positive portion of each coordinate ais. Taken as pairs, the aes determine three coordinate planes: the -plane, the -plane, and the -plane. These three coordinate planes separate the three-dimensional coordinate sstem into eight octants. The first octant is the one in which all three coordinates are positive. In this three-dimensional sstem, a point P in space is determined b an ordered triple,,, where,, and are as follows. directed distance from -plane to P directed distance from -plane to P directed distance from -plane to P -plane -plane (,, ) NASA -plane FIGURE. FIGURE. A three-dimensional coordinate sstem can have either a left-handed or a right-handed orientation. In this tet, ou will work eclusivel with righthanded sstems, as illustrated in Figure.. In a right-handed sstem, Octants II, III, and IV are found b rotating counterclockwise around the positive -ais. Octant V is verticall below Octant I. Octants VI, VII, and VIII are then found b rotating counterclockwise around the negative -ais. See Figure.. Octant I Octant II Octant III Octant IV Octant V Octant VI Octant VII Octant VIII FIGURE.

Section. The Three-Dimensional Coordinate Sstem 8 (,, ) 6 FIGURE. (,, ) (,, 0) (, 6, ) 6 (,, ) Eample Plotting Points in Space Plot each point in space. a.,, b., 6, c.,, 0 d.,, To plot the point,,, notice that,, and. To help visualie the point, locate the point, in the -plane (denoted b a cross in Figure.). The point,, lies three units above the cross. The other three points are also shown in Figure.. Now tr Eercise 5. The Distance and Midpoint Formulas Man of the formulas established for the two-dimensional coordinate sstem can be etended to three dimensions. For eample, to find the distance between two points in space, ou can use the Pthagorean Theorem twice, as shown in Figure.5. Note that a, b, and c. (,, ) (,, ) a b a + b (,, ) Distance Formula in Space The distance between the points,, and,, given b the Distance Formula in Space is d. (,, ) Eample Finding the Distance Between Two Points in Space (,, ) d = FIGURE.5 a+ b+ c a + b c (,, ) Find the distance between, 0, and,,. d 0 6 5 Now tr Eercise 7. Distance Formula in Space Substitute. Simplif. Notice the similarit between the Distance Formulas in the plane and in space. The Midpoint Formulas in the plane and in space are also similar. Midpoint Formula in Space The midpoint of the line segment joining the points,, and,, given b the Midpoint Formula in Space is,,.

8 Chapter Analtic Geometr in Three Dimensions Eample Using the Midpoint Formula in Space Find the midpoint of the line segment joining 5,, and 0,,. Using the Midpoint Formula in Space, the midpoint is 5 0,, 5,, 7 as shown in Figure.6. (5,, ) Midpoint: ( 5,, 7 ) (0,, ) FIGURE.6 FIGURE.7 (,, ) r ( h, k, j) Sphere: radius r; center ( h, k, j) Now tr Eercise. The Equation of a Sphere A sphere with center h, k, j and radius r is defined as the set of all points,, such that the distance between,, and h, k, j is r, as shown in Figure.7. Using the Distance Formula, this condition can be written as h k j r. B squaring each side of this equation, ou obtain the standard equation of a sphere. Standard Equation of a Sphere The standard equation of a sphere with center h, k, j and radius r is given b h k j r. Notice the similarit of this formula to the equation of a circle in the plane. h k j r Equation of sphere in space h k r Equation of circle in the plane As is true with the equation of a circle, the equation of a sphere is simplified when the center lies at the origin. In this case, the equation is r. Sphere with center at origin

Section. The Three-Dimensional Coordinate Sstem 85 Eample Finding the Equation of a Sphere Eploration Find the equation of the sphere that has the points,, 6 and,, as endpoints of a diameter. Eplain how this problem gives ou a chance to use all three formulas discussed so far in this section: the Distance Formula in Space, the Midpoint Formula in Space, and the standard equation of a sphere. Find the standard equation of the sphere with center,, and radius. Does this sphere intersect the -plane? h k j r Standard equation Substitute. From the graph shown in Figure.8, ou can see that the center of the sphere lies three units above the -plane. Because the sphere has a radius of, ou can conclude that it does intersect the -plane at the point,, 0. 5 5 FIGURE.8 r = (,, ) 6 7 (,, 0) Now tr Eercise 9. Eample 5 Finding the Center and Radius of a Sphere (,, ) 6 5 Find the center and radius of the sphere given b 6 8 0. To obtain the standard equation of this sphere, complete the square as follows. 6 8 0 6 8 6 9 8 9 6 r = 6 5 FIGURE.9 So, the center of the sphere is,,, and its radius is 6. See Figure.9. Now tr Eercise 9. Note in Eample 5 that the points satisfing the equation of the sphere are surface points, not interior points. In general, the collection of points satisfing an equation involving,, and is called a surface in space.

86 Chapter Analtic Geometr in Three Dimensions Finding the intersection of a surface with one of the three coordinate planes (or with a plane parallel to one of the three coordinate planes) helps one visualie the surface. Such an intersection is called a trace of the surface. For eample, the -trace of a surface consists of all points that are common to both the surface and the -plane. Similarl, the -trace of a surface consists of all points that are common to both the surface and the -plane. Eample 6 Finding a Trace of a Surface -trace: ( ) + ( ) = 8 6 0 8 0 Sphere: ( ) + ( ) + ( + ) = 5 FIGURE.0 Sketch the -trace of the sphere given b 5. To find the -trace of this surface, use the fact that ever point in the -plane has a -coordinate of ero. B substituting 0 into the original equation, the resulting equation will represent the intersection of the surface with the -plane. 5 Write original equation. 0 5 Substitute 0 for. 6 5 Simplif. 9 Subtract 6 from each side. Equation of circle You can see that the -trace is a circle of radius, as shown in Figure.0. Now tr Eercise 57. Technolog Most three-dimensional graphing utilities and computer algebra sstems represent surfaces b sketching several traces of the surface. The traces are usuall taken in equall spaced parallel planes. To graph an equation involving,, and with a three-dimensional function grapher, ou must first set the graphing mode to three-dimensional and solve the equation for. After entering the equation, ou need to specif a rectangular viewing cube (the three-dimensional analog of a viewing window). For instance, to graph the top half of the sphere from Eample 6, solve the equation for to obtain the solutions ± 5. The equation 5 represents the top half of the sphere. Enter this equation, as shown in Figure.. Net, use the viewing cube shown in Figure.. Finall, ou can displa the graph, as shown in Figure.. FIGURE. FIGURE. FIGURE.

Section. The Three-Dimensional Coordinate Sstem 87. Eercises VOCABULARY CHECK: Fill in the blanks.. A coordinate sstem can be formed b passing a -ais perpendicular to both the -ais and the -ais at the origin.. The three coordinate planes of a three-dimensional coordinate sstem are the, the, and the.. The coordinate planes of a three-dimensional coordinate sstem separate the coordinate sstem into eight.. The distance between the points,, and,, can be found using the in Space. 5. The midpoint of the line segment joining the points,, and,, given b the Midpoint Formula in Space is. 6. A is the set of all points,, such that the distance between,, and a fied point h, k, j is r. 7. A in is the collection of points satisfing an equation involving,, and. 8. The intersection of a surface with one of the three coordinate planes is called a of the surface. In Eercises and, approimate the coordinates of the points... B A C In Eercises 6, plot each point in the same three-dimensional coordinate sstem.. (a),,. (a), 0, 0 (b),, (b),, 5. (a),, 0 6. (a) 0,, (b),, (b), 0, In Eercises 7 0, find the coordinates of the point. 7. The point is located three units behind the -plane, three units to the right of the -plane, and four units above the -plane. 8. The point is located si units in front of the -plane, one unit to the left of the -plane, and one unit below the -plane. A B 5 5 6 C 9. The point is located on the -ais, 0 units in front of the -plane. 0. The point is located in the -plane, two units to the right of the -plane, and eight units above the -plane. In Eercises 6, determine the octant(s) in which is located so that the condition(s) is (are) satisfied.. > 0, < 0, > 0. < 0, > 0, < 0. > 0. < 0 5. < 0 6. > 0,, In Eercises 7, find the distance between the points. 7. 0, 0, 0, 5,, 6 8., 0, 0, 7, 0, 9.,, 5, 7,, 8 0.,, 9,,, 6.,,, 6, 0, 9.,, 7,,, 7. 0,, 0,, 0, 0.,, 0, 0, 6,

88 Chapter Analtic Geometr in Three Dimensions In Eercises 5 8, find the lengths of the sides of the right triangle. Show that these lengths satisf the Pthagorean Theorem. 5. 6. (0, 0, ) 7. 0, 0, 0,,,,,, 8., 0,,,,,, 0, In Eercises 9 and 0, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. 9.,,, 5,,,,, 0. 5,,, 7,,,, 5, In Eercises 8, find the midpoint of the line segment joining the points.. 0, 0, 0,,,., 5,,,,., 6, 0,,,., 5,,, 7, 5. 6,, 5,,, 6 6., 5, 5, 6,, 8 7., 8, 0, 7,, 8. 9, 5,, 9,, In Eercises 9 6, find the standard form of the equation of the sphere with the given characteristics. 9. 0. (,, ) 5 6 6 (, 5, ) (0,, 0) 5 6 r =. Center: 0,, ; radius:. Center:,, 8; radius: 6. Center:, 7, 5; diameter: 0. Center: 0, 5, 9; diameter: 8 5 6 5. Endpoints of a diameter:, 0, 0, 0, 0, 6 6. Endpoints of a diameter:,,,,, 6 (,, ) 6 (, 5, 0) (,, ) r = (,, ) In Eercises 7 56, find the center and radius of the sphere. 7. 5 0 8. 8 0 9. 6 0 0 50. 6 9 0 5. 8 9 0 5. 8 6 0 5. 9 9 9 8 6 7 7 0 5. 6 5 0 55. 9 9 9 6 8 0 56. 8 0 In Eercises 57 60, sketch the graph of the equation and sketch the specified trace. 57. 6; -trace 58. 5; -trace 59. 9; -trace 60. ; -trace In Eercises 6 and 6, use a three-dimensional graphing utilit to graph the sphere. 6. 6 8 0 6 0 6. 6 8 0 6. Crstals Crstals are classified according to their smmetr. Crstals shaped like cubes are classified as isometric. The vertices of an isometric crstal mapped onto a three-dimensional coordinate sstem are shown in the figure. Determine,,. (, 0, 0) (,, ) (0,, 0) (, 0, 8) FIGURE FOR 6 FIGURE FOR 6 6. Crstals Crstals shaped like rectangular prisms are classified as tetragonal. The vertices of a tetragonal crstal mapped onto a three-dimensional coordinate sstem are shown in the figure. Determine,,. 65. Architecture A spherical building has a diameter of 65 feet. The center of the building is placed at the origin of a three-dimensional coordinate sstem. What is the equation of the sphere? (, 0, 0) (,, ) (0,, 0)

Section. The Three-Dimensional Coordinate Sstem 89 Model It 66. Geograph Assume that Earth is a sphere with a radius of 96 miles. The center of Earth is placed at the origin of a three-dimensional coordinate sstem (see figure). Prime meridian 7. A sphere intersects the -plane. Describe the trace. 7. A plane intersects the -plane. Describe the trace. 7. A line segment has,, as one endpoint and m, m, m as its midpoint. Find the other endpoint,, of the line segment in terms of,,, m, m, and m. 7. Use the result of Eercise 7 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and the midpoint are, 0, and 5, 8, 7, respectivel. Skills Review Snthesis Equator (a) What is the equation of the sphere? (b) Lines of longitude that run north-south could be represented b what trace(s)? What shape would each of these traces form? (c) Lines of latitude that run east-west could be represented b what trace(s)? What shape would each of these traces form? (d) The prime meridian would represent what trace of the equation of the sphere from part (a)? (e) The equator would represent what trace of the equation of the sphere from part (a)? True or False? In Eercises 67 and 68, determine whether the statement is true or false. Justif our answer. 67. In the ordered triple,, that represents point P in space, is the directed distance from the -plane to P. 68. The surface consisting of all points,, in space that are the same distance r from the point h, j, k has a circle as its -trace. 69. Think About It What is the -coordinate of an point in the -plane? What is the -coordinate of an point in the -plane? What is the -coordinate of an point in the -plane? 70. Writing In two-dimensional coordinate geometr, the graph of the equation a b c 0 is a line. In three-dimensional coordinate geometr, what is the graph of the equation a b c 0? Is it a line? Eplain our reasoning. In Eercises 75 80, solve the quadratic equation b completing the square. 75. v v 0 76. 7 9 0 77. 5 5 0 78. 0 79. 9 0 80. 5 8 0 In Eercises 8 8, find the magnitude and direction angle of the vector v. 8. v i j 8. v i j 8. v i 5j 8. v 0i 7j In Eercises 85 and 86, find the dot product of u and v. 85. u, 86. u, 0 v, 5 v, 6 In Eercises 87 90, write the first five terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a linear model, a quadratic model, or neither. a 0 0 87. a 0 88. a n a n n a n a n 89. a 90. a a n a n a n a n n In Eercises 9 98, find the standard form of the equation of the conic with the given characteristics. 9. Circle: center: 5, ; radius: 7 9. Circle: center:, 6; radius: 9 9. Parabola: verte:, ; focus:, 9. Parabola: verte:, 5; focus:, 0 95. Ellipse: vertices: 0,, 6, ; minor ais of length 96. Ellipse: foci: 0, 0, 0, 6; major ais of length 9 97. Hperbola: vertices:, 0, 8, 0; foci: 0, 0,, 0 98. Hperbola: vertices:,,, 9; foci:, 0,, 0

80 Chapter Analtic Geometr in Three Dimensions. Vectors in Space What ou should learn Find the component forms of the unit vectors in the same direction of, the magnitudes of, the dot products of, and the angles between vectors in space. Determine whether vectors in space are parallel or orthogonal. Use vectors in space to solve real-life problems. Wh ou should learn it Vectors in space can be used to represent man phsical forces, such as tension in the cables used to support auditorium lights, as shown in Eercise on page 86. Vectors in Space Phsical forces and velocities are not confined to the plane, so it is natural to etend the concept of vectors from two-dimensional space to three-dimensional space. In space, vectors are denoted b ordered triples v v, v, v. Component form The ero vector is denoted b 0 0, 0, 0. Using the unit vectors i, 0, 0, j 0,, 0, and k 0, 0, in the direction of the positive -ais, the standard unit vector notation for v is v v i v j v k Unit vector form as shown in Figure.. If v is represented b the directed line segment from Pp, p, p to Qq, q, q, as shown in Figure.5, the component form of v is produced b subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point v v, v, v q p, q p, q p. 0, 0, k, 0, 0 i v, v, v j 0,, 0 P( p, p, p) v Q( q, q, q) FIGURE. FIGURE.5 SuperStock Vectors in Space. Two vectors are equal if and onl if their corresponding components are equal.. The magnitude (or length) of u u, u, u is u u u u.. A unit vector u in the direction of v is u v v,. The sum of u u, u, u and v v, v, v is u v u v, u v, u v. Vector addition 5. The scalar multiple of the real number c and u u, u, u is cu cu, cu, cu. Scalar multiplication 6. The dot product of u u, u, u and v v, v, v is u v u v u v u v. v 0. Dot product

Section. Vectors in Space 8 Eample Finding the Component Form of a Vector Find the component form and magnitude of the vector v having initial point,, and terminal point, 6,. Then find a unit vector in the direction of v. The component form of v is v, 6, 0,, which implies that its magnitude is v 0 8. The unit vector in the direction of v is u v v 0,, 0,, 0,,. Now tr Eercise. Technolog Some graphing utilities have the capabilit to perform vector operations, such as the dot product. Consult the user s guide for our graphing utilit for specific instructions. Eample Finding the Dot Product of Two Vectors Find the dot product of 0,, and,,. 0,,,, 0 0 6 6 Note that the dot product of two vectors is a real number, not a vector. Now tr Eercise 9. As was discussed in Section 6., the angle between two nonero vectors is the angle, 0, between their respective standard position vectors, as shown in Figure.6. This angle can be found using the dot product. (Note that the angle between the ero vector and another vector is not defined.) v u u θ Origin FIGURE.6 v Angle Between Two Vectors If is the angle between two nonero vectors u and v, then cos u v u v. If the dot product of two nonero vectors is ero, the angle between the vectors is 90 (recall that cos 90 0). Such vectors are called orthogonal. For instance, the standard unit vectors i, j, and k are orthogonal to each other.

8 Chapter Analtic Geometr in Three Dimensions Eample Finding the Angle Between Two Vectors θ 6.9 u =, 0, Find the angle between u, 0, and v,, 0. cos u v, 0,,, 0 u v, 0,,, 0 50 FIGURE.7 v =,, 0 This implies that the angle between the two vectors is arccos 50 as shown in Figure.7. 6.9 Now tr Eercise. Parallel Vectors u u = v w= v Recall from the definition of scalar multiplication that positive scalar multiples of a nonero vector v have the same direction as v, whereas negative multiples have the direction opposite that of v. In general, two nonero vectors u and v are parallel if there is some scalar c such that u cv. For eample, in Figure.8, the vectors u, v, and w are parallel because u v and w v. v Eample Parallel Vectors FIGURE.8 w Vector w has initial point,, 0 and terminal point,,. Which of the following vectors is parallel to w? a. u, 8, b. v, 8, Begin b writing w in component form. w,, 0,, a. Because u, 8,,, w, ou can conclude that u is parallel to w. b. In this case, ou need to find a scalar c such that, 8, c,,. However, equating corresponding components produces c for the first two components and c for the third. So, the equation has no solution, and the vectors v and w are not parallel. Now tr Eercise 7.

Section. Vectors in Space 8 You can use vectors to determine whether three points are collinear (lie on \ the same line). The points P, Q, and R are collinear if and onl if the vectors PQ and PR \ are parallel. Eample 5 Using Vectors to Determine Collinear Points Determine whether the points P,,, Q5,, 6, and R,, 0 are collinear. The component forms of PQ \ and PR \ are and PQ \ 5,, 6, 5, PR \,, 0 6, 0,. Because PR \ PQ \, ou can conclude that the are parallel. Therefore, the points P, Q, and R lie on the same line, as shown in Figure.9. R(,, 0) PR = 6, 0, PQ =, 5, P(,, ) Q(5,, 6) 0 8 6 FIGURE.9 Now tr Eercise. Eample 6 Finding the Terminal Point of a Vector The initial point of the vector v,, is P,, 6. What is the terminal point of this vector? Using the component form of the vector whose initial point is P,, 6 and whose terminal point is Qq, q, q, ou can write PQ \ q p, q p, q p q, q, q 6,,. This implies that q, q, and q 6. The solutions of these three equations are q 7, q, and q 5. So, the terminal point is Q7,, 5. Now tr Eercise 5.

8 Chapter Analtic Geometr in Three Dimensions Application In Section 6., ou saw how to use vectors to solve an equilibrium problem in a plane. The net eample shows how to use vectors to solve an equilibrium problem in space. P(, 0, 0) u FIGURE.0 R(, 0, 0) Q(0,, 0) v S(0,, ) w Eample 7 Solving an Equilibrium Problem A weight of 80 pounds is supported b three ropes. As shown in Figure.0, the weight is located at S0,,. The ropes are tied to the points P, 0, 0, Q0,, 0, and R, 0, 0. Find the force (or tension) on each rope. The (downward) force of the weight is represented b the vector w 0, 0, 80. The force vectors corresponding to the ropes are as follows. 0, 0, 0 u u SP\ u SP \ u,, 0,, 0 v v SQ\ v0 SQ \ v 5 0, 5, 5 0, 0, 0 SR\ SR \,, For the sstem to be in equilibrium, it must be true that u v w 0 or u v w. This ields the following sstem of linear equations. u 000 u 5 v 000 u 5 v 80 Using the techniques demonstrated in Chapter 7, ou can find the solution of the sstem to be u 60.0 v 56.7 60.0. So, the rope attached at point P has 60 pounds of tension, the rope attached at point Q has about 56.7 pounds of tension, and the rope attached at point R has 60 pounds of tension. Now tr Eercise.

Section. Vectors in Space 85. Eercises VOCABULARY CHECK: Fill in the blanks.. The vector is denoted b 0 0, 0, 0.. The standard unit vector notation for a vector v is given b.. The of a vector v is produced b subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point.. If the dot product of two nonero vectors is ero, the angle between the vectors is 90 and the vectors are called. 5. Two nonero vectors u and v are if there is some scalar c such that u cv. In Eercises and, (a) find the component form of the vector v and (b) sketch the vector with its initial point at the origin... (, 0, ) In Eercises and, (a) write the component form of the vector v, (b) find the magnitude of v, and (c) find a unit vector in the direction of v.. Initial point: 6,, Terminal point:,,. Initial point: 7,, 5 Terminal point: 0, 0, In Eercises 5 and 6, sketch each scalar multiple of v. 5. v,, (a) (b) (c) v (d) 0v 6. v,, (a) (b) (c) (d) In Eercises 7 0, find the vector, given and w < > 5, 0, 5. v <,, >, u <,, >, 7. u v 8. 7u v 5 w 9. u w 0. u v 0 (0,, ) v v v (,, 0) 5 v (,, ) In Eercises 6, find the magnitude of v.. v 7, 8, 7. v, 0, 5. v i j 7k. v i j 6k 5. Initial point:,, Terminal point:, 0, 6. Initial point: 0,, 0 Terminal point:,, In Eercises 7 and 8, find a unit vector (a) in the direction of u and (b) in the direction opposite of u. 7. u 8i j k 8. u i 5j 0k In Eercises 9, find the dot product of u and v. 9. u,, v, 5, 8 0. u,, 6 v, 0,. u i 5j k v 9i j k. u j 6k v 6i j k In Eercises 6, find the angle between the vectors.. u 0,,. u,, 0 v, 0, v,, 5. u 0i 0j 6. u 8j 0k v j 8k v 0i 5k

86 Chapter Analtic Geometr in Three Dimensions In Eercises 7 0, determine whether u and v are orthogonal, parallel, or neither. 7. u, 6, 5 8. u,, v 8,, 0 v,, 5 9. u i j k 0. u i j k v i 0j k v 8i j 8k In Eercises, use vectors to determine whether the points are collinear.. 5,,, 7,,,, 5,., 7,,, 8,, 0, 6, 7.,,,,, 5,,,. 0,,,, 5, 6,, 6, 7 In Eercises 5 8, the vector v and its initial point are given. Find the terminal point. 5. v,, 7 6. v,, Initial point:, 5, 0 Initial point: 6,, 7., 8. v 5,, Initial point: Initial point:,, 9. Determine the values of c such that cu, where u i j k. 0. Determine the values of c such that cu, where u i j k. In Eercises and, write the component form of v.. v lies in the -plane, has magnitude, and makes an angle of 5 with the positive -ais.. v lies in the -plane, has magnitude 0, and makes an angle of 60 with the positive -ais.. Tension The weight of a crate is 500 newtons. Find the tension in each of the supporting cables shown in the figure. 5 cm 65 cm D A C 70 cm B 60 cm 5 cm,,. Tension The lights in an auditorium are -pound disks of radius 8 inches. Each disk is supported b three equall spaced cables that are L inches long (see figure). (a) Write the tension T in each cable as a function of L. Determine the domain of the function. (b) Use the function from part (a) to complete the table. (c) Use a graphing utilit to graph the function in part (a). What are the asmptotes of the graph? Interpret their meaning in the contet of the problem. (d) Determine the minimum length of each cable if a cable can carr a maimum load of 0 pounds. Snthesis True or False? In Eercises 5 and 6, determine whether the statement is true or false. Justif our answer. 5. If the dot product of two nonero vectors is ero, then the angle between the vectors is a right angle. 6. If AB and AC are parallel vectors, then points A, B, and C are collinear. 7. What is known about the nonero vectors u and v if u v < 0? Eplain. 8. Writing Consider the two nonero vectors u and v. Describe the geometric figure generated b the terminal points of the vectors tv, u tv, and su tv, where s and t represent real numbers. Skills Review Model It L 8 in. L 0 5 0 5 0 5 50 T In Eercises 9 5, find a set of parametric equations for the rectangular equation using (a) t and (b) t. 9. 50. 5. 8 5.

Section. The Cross Product of Two Vectors 87. The Cross Product of Two Vectors What ou should learn Find cross products of vectors in space. Use geometric properties of cross products of vectors in space. Use triple scalar products to find volumes of parallelepipeds. Wh ou should learn it The cross product of two vectors in space has man applications in phsics and engineering. For instance, in Eercise on page 8, the cross product is used to find the torque on the crank of a biccle s brake. Carl Schneider/Gett Images The Cross Product Man applications in phsics, engineering, and geometr involve finding a vector in space that is orthogonal to two given vectors. In this section, ou will stud a product that will ield such a vector. It is called the cross product, and it is convenientl defined and calculated using the standard unit vector form. Definition of Cross Product of Two Vectors in Space Let u u i u j u k It is important to note that this definition applies onl to three-dimensional vectors. The cross product is not defined for two-dimensional vectors. A convenient wa to calculate u v is to use the following determinant form with cofactor epansion. (This determinant form is used simpl to help remember the formula for the cross product it is technicall not a determinant because the entries of the corresponding matri are not all real numbers.) i j k u v u u u Put u in Row. v v v Put v in Row. u v u v i u v u v j u v u v k u v u v i u v u v j u v u v k Note the minus sign in front of the j-component. Recall from Section 8. that each of the three determinants can be evaluated b using the following pattern. a a b b a b a b and v v i v j v k be vectors in space. The cross product of u and v is the vector u v u v u v i u v u v j u v u v k. Eploration Find each cross product. What can ou conclude? a. i j b. i k c. j k

88 Chapter Analtic Geometr in Three Dimensions Eample Finding Cross Products Technolog Some graphing utilities have the capabilit to perform vector operations, such as the cross product. Consult the user s guide for our graphing utilit for specific instructions. Eploration Calculate u v and v u for several values of u and v. What do our results impl? Interpret our results geometricall. Given u i j k and v i j k, find each cross product. a. u v b. v u c. v v a. u v b. c. v u i j i i j 6k i j 5k i j k k i j j i j 6 k i j 5k i j k v v 0 Now tr Eercise 9. k k The results obtained in Eample suggest some interesting algebraic properties of the cross product. For instance, u v v u and v v 0. These properties, and several others, are summaried in the following list. Algebraic Properties of the Cross Product Let u, v, and w be vectors in space and let c be a scalar.. u v v u. u v w u v u w. cu v cu v u cv. u 0 0 u 0 5. u u 0 6. u v w u v w For proofs of the Algebraic Properties of the Cross Product, see Proofs in Mathematics on page 87.

Section. The Cross Product of Two Vectors 89 Geometric Properties of the Cross Product The first propert listed on the preceding page indicates that the cross product is not commutative. In particular, this propert indicates that the vectors u v and v u have equal lengths but opposite directions. The following list gives some other geometric properties of the cross product of two vectors. Geometric Properties of the Cross Product Let u and v be nonero vectors in space, and let be the angle between u and v.. u v is orthogonal to both u and v.. u v u v sin k = i j. u v 0 if and onl if u and v are scalar multiples of each other.. u v area of parallelogram having u and v as adjacent sides. i FIGURE. j -plane For proofs of the Geometric Properties of the Cross Product, see Proofs in Mathematics on page 88. Both u v and v u are perpendicular to the plane determined b u and v. One wa to remember the orientations of the vectors u, v, and u v is to compare them with the unit vectors i, j, and k i j, as shown in Figure.. The three vectors u, v, and u v form a right-handed sstem. Eample Using the Cross Product Find a unit vector that is orthogonal to both u i j k and v i 6j. 8 6 ( 6,, 6) The cross product u v, as shown in Figure., is orthogonal to both u and v. 0 i j k u v 6 (,, ) u FIGURE. u v 6 6 (, 6, 0) v Because 6i j 6k u v 6 6 8 9 a unit vector orthogonal to both u and v is u v u v i j k. Now tr Eercise 5.

80 Chapter Analtic Geometr in Three Dimensions In Eample, note that ou could have used the cross product v u to form a unit vector that is orthogonal to both u and v. With that choice, ou would have obtained the negative of the unit vector found in the eample. The fourth geometric propert of the cross product states that u v is the area of the parallelogram that has u and v as adjacent sides. A simple eample of this is given b the unit square with adjacent sides of i and j. Because i j k and k, it follows that the square has an area of. This geometric propert of the cross product is illustrated further in the net eample. Eample Geometric Application of the Cross Product D (5, 0, 6) 8 FIGURE. 8 6 Eploration C (,, 7) B (, 6, ) 6 A (5,, 0) 8 If ou connect the terminal points of two vectors u and v that have the same initial points, a triangle is formed. Is it possible to use the cross product u v to determine the area of the triangle? Eplain. Verif our conclusion using two vectors from Eample. Show that the quadrilateral with vertices at the following points is a parallelogram. Then find the area of the parallelogram. Is the parallelogram a rectangle? A5,, 0, B, 6,, C,, 7, D5, 0, 6 From Figure. ou can see that the sides of the quadrilateral correspond to the following four vectors. \ Because CD \ AB \ and CB \ AD \, ou can conclude that AB is parallel to CD \ and AD \ is parallel to CB \. It follows that the quadrilateral is a parallelogram with AB \ and AD \ as adjacent sides. Moreover, because 6 i j k AB \ AD \ 6i 8j 6k 0 the area of the parallelogram is You can tell whether the parallelogram is a rectangle b finding the angle between the vectors AB \ and AD \. Because AB \ i j k CD \ i j k AB \ AD \ 0i j 6k CB \ 0i j 6k AD \ AB \ AD \ 6 8 6 06.9. sin AB\ AD \ AB \ AD \ 06 60 arcsin 0.998 86. 90, 0.998 the parallelogram is not a rectangle. Now tr Eercise 7.

The Triple Scalar Product Section. The Cross Product of Two Vectors 8 For the vectors u, v, and w in space, the dot product of u and v w is called the triple scalar product of u, v, and w. proj v w u v w u w v Area of base v w Volume of parallelepiped u v w FIGURE. The Triple Scalar Product For u u i u j u and the triple scalar product is given b k, v v i v j v k, w w i w j w k, u u u u v w v v v w w w. If the vectors u, v, and w do not lie in the same plane, the triple scalar product u v w can be used to determine the volume of the parallelepiped (a polhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges, as shown in Figure.. Geometric Propert of Triple Scalar Product The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is given b V u v w. Eample Volume b the Triple Scalar Product (, 5, ) u FIGURE.5 6 (,, ) w v (0,, ) Find the volume of the parallelepiped having u i 5j k, as adjacent edges, as shown in Figure.5. The value of the triple scalar product is 5 u v w 0 6. So, the volume of the parallelepiped is u v w 6 v j k, 5 0 56 6 6. Now tr Eercise 9. and 0 w i j k

8 Chapter Analtic Geometr in Three Dimensions. Eercises VOCABULARY CHECK: Fill in the blanks.. To find a vector in space that is orthogonal to two given vectors, find the of the two vectors.. u u. u v. The dot product of u and v w is called the of u, v, and w. In Eercises, find the cross product of the unit vectors and sketch the result.. j i. k j. i k. k i In Eercises 5, find u v and show that it is orthogonal to both u and v. 5. u,, 5 6. u 6, 8, v 0,, v,, 7. u 0, 0, 6 8. u 5, 5, v 7, 0, 0 v,, 9. u 6i j k 0. u i j 5 k v i j k v i j k.. u i v i j k v j k. u i k. u i k v j k In Eercises 5 0, find a unit vector orthogonal to u and v. 5. u i j 6. u i j v j k v i k 7. u i j 5k 8. u 7i j 5k v i j 0 k v i 8j 5k 9. u i j k 0. u i j k v i j k In Eercises 6, find the area of the parallelogram that has the vectors as adjacent sides.. u k. u i j k v i k v i k. u i j 6k. u i j k v i j 5k v i j k 5. u,, 6. u,, v 0,, v j k v i j k v 5, 0, In Eercises 7 and 8, (a) verif that the points are the vertices of a parallelogram, (b) find its area, and (c) determine whether the parallelogram is a rectangle. 7. A,,, B,,, C0, 5, 6, D,, 8 8. A,,, B,,, C6, 5,, D7, 7, 5 In Eercises 9, find the area of the triangle with the given vertices. (The area A of the triangle having u and v as adjacent sides is given b A u v. ) 9. 0, 0, 0,,,,, 0, 0 0.,,,, 0,,,, 0.,, 5,,, 0,, 0, 6.,, 0,,, 0, 0, 0, In Eercises 6, find the triple scalar product... 5. 6. In Eercises 7 0, use the triple scalar product to find the volume of the parallelepiped having adjacent edges u, v, and w. 7. u i j 8. u i j k (, 0, ) u,,, v,, 0, w 0, 0, u, 0,, v 0,, 0, w 0, 0, u i j k, v i j, w i j k u i j 7k, v i k, w j 6k v j k w i k w u v (0,, ) (,, 0) (,, ) w (, 0, ) v j k w i k u v (0,, )

Section. The Cross Product of Two Vectors 8 9. u 0,, 0. u,, In Eercises and, find the volume of the parallelepiped with the given vertices... v 0, 0, w, 0, (, 0, ) w v (0,, ) u (0, 0, ) (, 0, ) A0, 0, 0, B, 0, 0, C,,, D0,,, E, 5,, F0, 5,, G0,, 6, H,, 6 A0, 0, 0, B,, 0, C, 0,, D0,,, E,,, F,,, G,,, H,,. Torque Both the magnitude and direction of the force on a crankshaft change as the crankshaft rotates. Use the technique given in Eercise to find the magnitude of the torque on the crankshaft using the position and data shown in the figure. Model It v,, w, 0, w u (,, ) v (,, ). Torque The brakes on a biccle are applied b using a downward force of p pounds on the pedal when the si-inch crank makes a 0 angle with the horiontal (see figure). Vectors representing the position of the crank and the force are V cos 0j sin 0k and F pk, respectivel. 6 in. V 0 F = p lb (a) The magnitude of the torque on the crank is given b V F. Using the given information, write the torque T on the crank as a function of p. (b) Use the function from part (a) to complete the table. p 5 0 5 0 5 0 5 T Snthesis FIGURE FOR True or False? In Eercises 5 and 6, determine whether the statement is true or false. Justif our answer. 5. The cross product is not defined for vectors in the plane. 6. If u and v are vectors in space that are nonero and nonparallel, then u v v u. 7. Think About It If the magnitudes of two vectors are doubled, how will the magnitude of the cross product of the vectors change? 8. Proof Consider the vectors u cos, sin, 0 and v cos, sin, 0, where >. Find the cross product of the vectors and use the result to prove the identit sin Skills Review 0.6 ft V 60 F = 000 lb sin cos cos sin. In Eercises 9 56, evaluate the epression without using a calculator. 9. cos 80 50. tan 00 5. sin 690 5. cos 90 5. sin 9 6 5. 55. tan 5 56. cos 7 6 tan 0 In Eercises 57 and 58, sketch the region determined b the constraints. Then find the minimum and maimum values of the objective function and where the occur, subject to the indicated constraints. 57. Objective function: 58. Objective function: 6 6 7 Constraints: Constraints: 6 0 6 0 5 0 0 0 0

8 Chapter Analtic Geometr in Three Dimensions. Lines and Planes in Space What ou should learn Find parametric and smmetric equations of lines in space. Find equations of planes in space. Sketch planes in space. Find distances between points and planes in space. Wh ou should learn it Equations in three variables can be used to model real-life data. For instance, in Eercise 7 on page 8, ou will determine how changes in the consumption of two tpes of milk affect the consumption of a third tpe of milk. Lines in Space In the plane, slope is used to determine an equation of a line. In space, it is more convenient to use vectors to determine the equation of a line. In Figure.6, consider the line L through the point P,, and parallel to the vector v a, b, c. P(,, ) Direction vector for L Q(,, ) v = a, b, c PQ = tv L FIGURE.6 Mark E. Gibson/Corbis The vector v is the direction vector for the line L, and a, b, and c are the direction numbers. One wa of describing the line L is to sa that it consists of all points Q,, for which the vector PQ \ is parallel to v. This means that PQ \ is a scalar multiple of v, and ou can write PQ \ tv, where t is a scalar. PQ \,, at, bt, ct tv B equating corresponding components, ou can obtain the parametric equations of a line in space. Parametric Equations of a Line in Space A line L parallel to the vector v a, b, c and passing through the point P,, is represented b the parametric equations at, bt, and ct. If the direction numbers a, b, and c are all nonero, ou can eliminate the parameter t to obtain the smmetric equations of a line. a b c Smmetric equations

(,, ) 6 FIGURE.7 L 6 6 v =,, Eample Finding Parametric and Smmetric Equations Find parametric and smmetric equations of the line L that passes through the point,, and is parallel to v,,. To find a set of parametric equations of the line, use the coordinates,, and and direction numbers a, b, and c (see Figure.7). t, Because a, b, and c are all nonero, a set of smmetric equations is Now tr Eercise. Parametric equations Smmetric equations Neither the parametric equations nor the smmetric equations of a given line are unique. For instance, in Eample, b letting t in the parametric equations ou would obtain the point,, 0. Using this point with the direction numbers a, b, and c produces the parametric equations t, t,. t, Section. Lines and Planes in Space 85 and t t. To check the answer to Eample, verif that the two original points lie on the line. To see this, substitute t 0 and t in the parametric equations as follows. t 0: 0 0 50 0 t : 5 5 Eample Parametric and Smmetric Equations of a Line Through Two Points Find a set of parametric and smmetric equations of the line that passes through the points,, 0 and,, 5. Begin b letting P,, 0 and Q,, 5. Then a direction vector for the line passing through P and Q is v PQ \,, 5 0,, 5 a, b, c. Using the direction numbers a, b, and c 5 with the initial point P,, 0, ou can obtain the parametric equations t, and Because a, b, and c are all nonero, a set of smmetric equations is 5. t, Now tr Eercise 7. 5t. Parametric equations Smmetric equations

86 Chapter Analtic Geometr in Three Dimensions Planes in Space You have seen how an equation of a line in space can be obtained from a point on the line and a vector parallel to it. You will now see that an equation of a plane in space can be obtained from a point in the plane and a vector normal (perpendicular) to the plane. P n Q n PQ = 0 FIGURE.8 Consider the plane containing the point P,, having a nonero normal vector n a, b, c, as shown in Figure.8. This plane consists of all points Q,, for which the vector PQ \ is orthogonal to n. Using the dot product, ou can write the following. n PQ \ 0 a, b, c,, 0 a b c 0 PQ \ is orthogonal to n. The third equation of the plane is said to be in standard form. Standard Equation of a Plane in Space The plane containing the point,, and having normal vector n a, b, c can be represented b the standard form of the equation of a plane a b c 0. Regrouping terms ields the general form of the equation of a plane in space a b c d 0. General form of equation of plane Given the general form of the equation of a plane, it is eas to find a normal vector to the plane. Use the coefficients of,, and to write n a, b, c. Eploration Consider the following four planes. 6 5 6 9 What are the normal vectors for each plane? What can ou sa about the relative positions of these planes in space?

Section. Lines and Planes in Space 87 Eample Finding an Equation of a Plane in Three-Space 5 (,, ) v (,, ) u (0,, ) FIGURE.9 Find the general form of the equation of the plane passing through the points,,, 0,,, and,,. To find the equation of the plane, ou need a point in the plane and a vector that is normal to the plane. There are three choices for the point, but no normal vector is given. To obtain a normal vector, use the cross product of vectors u and v etending from the point,, to the points 0,, and,,, as shown in Figure.9. The component forms of u and v are u 0,,,, 0 v,,, 0, and it follows that i j k n u v 0 0 9i 6j k a, b, c is normal to the given plane. Using the direction numbers for n and the initial point,,,,, ou can determine an equation of the plane to be a b c 0 9 6 0 9 6 6 0 0. Standard form General form Check that each of the three points satisfies the equation 0. Now tr Eercise 5. n θ n θ Two distinct planes in three-space either are parallel or intersect in a line. If the intersect, ou can determine the angle 90 between them from the angle between their normal vectors, as shown in Figure.0. Specificall, if vectors n and n are normal to two intersecting planes, the angle between the normal vectors is equal to the angle between the two planes and is given b cos n n n n. 0 Angle between two planes FIGURE.0 Consequentl, two planes with normal vectors n and n are. perpendicular if n n 0.. parallel if n is a scalar multiple of n.

88 Chapter Analtic Geometr in Three Dimensions Eample Finding the Line of Intersection of Two Planes Plane FIGURE. θ 5.55 Line of Intersection Plane Find the angle between the two planes given b 0 Equation for plane 0 Equation for plane and find parametric equations of their line of intersection (see Figure.). The normal vectors for the planes are n,, and n,,. Consequentl, the angle between the two planes is determined as follows. cos n n 6 6 0.5909. n n 67 0 This implies that the angle between the two planes is You can find the line of intersection of the two planes b simultaneousl solving the two linear equations representing the planes. One wa to do this is to multipl the first equation b and add the result to the second equation. 0 0 Substituting 7 back into one of the original equations, ou can determine that 7. Finall, b letting t 7, ou obtain the parametric equations t at, t bt, Because,, 0, 0, 0 lies in both planes, ou can substitute for,, and in these parametric equations, which indicates that a, b, and c 7 are direction numbers for the line of intersection. Now tr Eercise. 0 0 7 0 5.55. 7t ct. 7 Note that the direction numbers in Eample can also be obtained from the cross product of the two normal vectors as follows. i j k n n i j k i j 7k This means that the line of intersection of the two planes is parallel to the cross product of their normal vectors.

Section. Lines and Planes in Space 89 Technolog Most three-dimensional graphing utilities and computer algebra sstems can graph a plane in space. Consult the user s guide for our utilit for specific instructions. Sketching Planes in Space As discussed in Section., if a plane in space intersects one of the coordinate planes, the line of intersection is called the trace of the given plane in the coordinate plane. To sketch a plane in space, it is helpful to find its points of intersection with the coordinate aes and its traces in the coordinate planes. For eample, consider the plane. Equation of plane You can find the -trace b letting 0 and sketching the line -trace in the -plane. This line intersects the -ais at, 0, 0 and the -ais at 0, 6, 0. In Figure., this process is continued b finding the -trace and the -trace and then shading the triangular region ling in the first octant. 6 6 6 (, 0, 0) (0, 0, ) (0, 6, 0) (, 0, 0) (, 0, 0) (0, 6, 0) (0, 0, ) (0, 6, 0) 6 6 6 6 6 6 (a) -trace ( 0): (b) -trace 0: (c) -trace 0: FIGURE. (0, 0, ) If the equation of a plane has a missing variable, such as, the plane must be parallel to the ais represented b the missing variable, as shown in Figure.. If two variables are missing from the equation of a plane, then it is parallel to the coordinate plane represented b the missing variables, as shown in Figure.. (, 0, 0) ( 0, 0, d ) c Plane: + = Plane is parallel to -ais FIGURE. ( d d ) ( 0,, 0 a, 0, 0 ) b (a) Plane a d 0 (b) Plane b d 0 (c) Plane c d 0 is parallel to -plane. is parallel to -plane. is parallel to -plane. FIGURE.

80 Chapter Analtic Geometr in Three Dimensions n proj n PQ D proj n PQ \ FIGURE.5 P Q D Distance Between a Point and a Plane The distance D between a point Q and a plane is the length of the shortest line segment connecting Q to the plane, as shown in Figure.5. If P is an point in the plane, ou can find this distance b projecting the vector PQ \ onto the normal vector n. The length of this projection is the desired distance. Distance Between a Point and a Plane The distance between a plane and a point Q (not in the plane) is D proj n PQ \ PQ\ n n where P is a point in the plane and n is normal to the plane. To find a point in the plane given b a b c d 0, where a 0, let 0 and 0. Then, from the equation a d 0, ou can conclude that the point da, 0, 0 lies in the plane. Eample 5 Finding the Distance Between a Point and a Plane Find the distance between the point Q, 5, and the plane 6. You know that n,, is normal to the given plane. To find a point in the plane, let 0 and 0, and obtain the point P, 0, 0. The vector from P to Q is PQ \, 5 0, 0, 5,. The formula for the distance between a point and a plane produces D PQ\ n, 5,,, 9 5 8 6. n Now tr Eercise 5. The choice of the point P in Eample 5 is arbitrar. Tr choosing a different point to verif that ou obtain the same distance.

Section. Lines and Planes in Space 8. Eercises VOCABULARY CHECK: Fill in the blanks.. The vector for a line L is given b v.. The of a line in space are given b at, bt, and ct.. If the direction numbers a, b, and c of the vector v a, b, c are all nonero, ou can eliminate the parameter to obtain the of a line.. A vector that is perpendicular to a plane is called. 5. The standard form of the equation of a plane is given b. In Eercises 6, find a set of (a) parametric equations and (b) smmetric equations for the line through the point and parallel to the specified vector or line. (For each line, write the direction numbers as integers.) v Point. 0, 0, 0., 5,.,, 0. 5, 0, 0 5.,, 5 6., 0, Parallel to In Eercises 7, find (a) a set of parametric equations and (b) if possible, a set of smmetric equations of the line that passes through the given points. (For each line, write the direction numbers as integers.) 7., 0,,,, 8.,, 0, 0, 8, 9., 8, 5,,, 6 0.,,,, 5,.,,,,, 5.,, 5,,,..,,,, 5,,,,,, 0 In Eercises 5 and 6, sketch a graph of the line. 5. t, t, 6. 5 t, t, t In Eercises 7, find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point 7.,, 8., 0, 9. 5, 6, v,,, 7, 0 v i j k v i k 5 t, 7 t, t t, 5 t, 7 t Perpendicular to n i n k n i j k 5 t Point 0. 0, 0, 0., 0, 0. 0, 0, 6 Perpendicular to In Eercises 6, find the general form of the equation of the plane passing through the three points.. 0, 0, 0,,,,,,.,,,, 5,,,, 5.,,,,,,,, 0 6. 5,,,,,,,, In Eercises 7 and 8, find the general form of the equation of the plane with the given characteristics. 7. The plane passes through the point, 5, and is parallel to the -plane. 8. The plane passes through the points,, and,, and is perpendicular to the plane. In Eercises 9, determine whether the planes are parallel, orthogonal, or neither. If the are neither parallel nor orthogonal, find the angle of intersection. 9. 5 0. 7 9. 8. 5 5 5 8 0 n j 5k t, t, t t, t, t 5 5 5 In Eercises 6, (a) find the angle between the two planes and (b) find parametric equations of their line of intersection.. 5 6. 5 0

8 Chapter Analtic Geometr in Three Dimensions 5. 0 6. 5 In Eercises 7, plot the intercepts and sketch a graph of the plane. 7. 6 8. 9. 0. 5. 6. 6 In Eercises 6, find the distance between the point and the plane.. 0, 0, 0.,, 8 8 5.,, 6.,, 5 Model It 6 0 7. Data Analsis: Milk Consumption The table shows the per capita milk consumptions (in gallons) of different tpes of plain milk in the United States from 999 to 00. Consumption of light and skim milks, reduced-fat milk, and whole milk are represented b the variables,, and, respectivel. (Source: U.S. Department of Agriculture) Year 999 6. 7. 7.8 000 6. 7. 7.7 00 5.9 7.0 7. 00 5.7 7.0 7. 00 5.6 6.9 7. A model for the data is given b 0.8 0.6 0.. (a) Complete a fifth column in the table using the model to approimate for the given values of and. (b) Compare the approimations from part (a) with the actual values of. (c) According to this model, an increases or decreases in consumption of two tpes of milk will have what effect on the consumption of the third tpe of milk? 8. Mechanical Design A chute at the top of a grain elevator of a combine funnels the grain into a bin as shown in the figure. Find the angle between two adjacent sides. Snthesis True or False? In Eercises 9 and 50, determine whether the statement is true or false. Justif our answer. 9. Ever two lines in space are either intersecting or parallel. 50. Two nonparallel planes in space will alwas intersect. 5. The direction numbers of two distinct lines in space are 0, 8, 0, and 5, 7, 0. What is the relationship between the lines? Eplain. 5. Eploration (a) Describe and find an equation for the surface generated b all points,, that are two units from the point,,. (b) Describe and find an equation for the surface generated b all points,, that are two units from the plane 0. Skills Review T (,, 8) P (6, 0, 0) In Eercises 5 56, convert the polar equation to rectangular form. 5. r 0 5. 55. r cos 56. r cos In Eercises 57 60, convert the rectangular equation to polar form. 57. 9 58. 0 59. 5 60. 0 S (0, 0, 0) R (7, 7, 8) Q (6, 6, 0)