🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Precalculus with Limits ## Ron Larson ## Chapter 11 ## Analytic Geometry in Three Dimensions ## Educators  ### Problem 1 A _______ coordinate system can be formed by passing a$z$-axis perpendicular to both the$x$-axis and the$y$-axis at the origin. Ernest C. Numerade Educator ### Problem 2 The three coordinate planes of a three-dimensional coordinate system are the _______, the _______, and the _______ . Taylor S. Numerade Educator ### Problem 3 The coordinate planes of a three-dimensional coordinate system separate the coordinate system into eight _______. Ernest C. Numerade Educator ### Problem 4 The distance between the points$(x_1, y_1, z_1)$and$(x_2, y_2, z_2)$can be found using the _______ _______ in Space. Taylor S. Numerade Educator ### Problem 5 The midpoint of the line segment joining the points$(x_1, y_1, z_1)$and$(x_2, y_2, z_2)$given by the Midpoint Formula in Space is _______ . Ernest C. Numerade Educator ### Problem 6 A _______ is the set of all points$(x, y, z)$such that the distance between$(x, y, z)$and a fixed point$(h, k, j)$is$r$. Taylor S. Numerade Educator ### Problem 7 A _______ in _______ is the collection of points satisfying an equation involving$x$,$y$, and$z$. Ernest C. Numerade Educator ### Problem 8 The intersection of a surface with one of the three coordinate planes is called a _______ of the surface. Taylor S. Numerade Educator ### Problem 9 In Exercises 9 and 10, approximate the coordinates of the points. Ernest C. Numerade Educator ### Problem 10 In Exercises 9 and 10, approximate the coordinates of the points. Taylor S. Numerade Educator ### Problem 11 In Exercises 11-16, plot each point in the same three-dimensional coordinate system. (a)$(2, 1, 3)$(b)$(1, -1, -2)$ Ernest C. Numerade Educator ### Problem 12 In Exercises 11-16, plot each point in the same three-dimensional coordinate system. (a)$(3, 0, 0)$(b)$(-3, -2, -1)$ Taylor S. Numerade Educator ### Problem 13 In Exercises 11-16, plot each point in the same three-dimensional coordinate system. (a)$(3, -1, 0)$(b)$(-4, 2, 2)$ Ernest C. Numerade Educator ### Problem 14 In Exercises 11-16, plot each point in the same three-dimensional coordinate system. (a)$(0, 4, -3)$(b)$(4, 0, 4)$ Taylor S. Numerade Educator ### Problem 15 In Exercises 11-16, plot each point in the same three-dimensional coordinate system. (a)$(3, -2, 5)$(b)$(\frac{3}{2}, 4, -2)$ Ernest C. Numerade Educator ### Problem 16 In Exercises 11-16, plot each point in the same three-dimensional coordinate system. (a)$(5, -2, 2)$(b)$(5, -2, -2)$ Taylor S. Numerade Educator ### Problem 17 In Exercises 17-20, find the coordinates of the point. The point is located three units behind the$yz$-plane, four units to the right of the$xz$-plane, and five units above the$xy$-plane. Ernest C. Numerade Educator ### Problem 18 In Exercises 17-20, find the coordinates of the point. The point is located seven units in front of the$yz$-plane,two units to the left of the$xz$-plane, and one unit below the$xy$-plane. Taylor S. Numerade Educator ### Problem 19 In Exercises 17-20, find the coordinates of the point. The point is located on the$x$-axis, eight units in front of the$yz$-plane. Ernest C. Numerade Educator ### Problem 20 In Exercises 17-20, find the coordinates of the point. The point is located in the$yz$-plane, one unit to the right of the$xz$-plane, and six units above the$xy$-plane. Taylor S. Numerade Educator ### Problem 21 In Exercises 21-26, determine the octant(s) in which$(x, y, z)$is located so that the condition(s) is (are) satisfied.$x>0, y<0, z>0$ Ernest C. Numerade Educator ### Problem 22 In Exercises 21-26, determine the octant(s) in which$(x, y, z)$is located so that the condition(s) is (are) satisfied.$x<0, y>0, z<0$ Taylor S. Numerade Educator ### Problem 23 In Exercises 21-26, determine the octant(s) in which$(x, y, z)$is located so that the condition(s) is (are) satisfied.$z>0$ Ernest C. Numerade Educator ### Problem 24 In Exercises 21-26, determine the octant(s) in which$(x, y, z)$is located so that the condition(s) is (are) satisfied.$y<0$ Taylor S. Numerade Educator ### Problem 25 In Exercises 21-26, determine the octant(s) in which$(x, y, z)$is located so that the condition(s) is (are) satisfied.$xy<0$ Ernest C. Numerade Educator ### Problem 26 In Exercises 21-26, determine the octant(s) in which$(x, y, z)$is located so that the condition(s) is (are) satisfied.$yz<0$ Taylor S. Numerade Educator ### Problem 27 In Exercises 27-36, find the distance between the points.$(0, 0, 0), \quad (5, 2, 6)$ Ernest C. Numerade Educator ### Problem 28 In Exercises 27-36, find the distance between the points.$(1, 0, 0), \quad (7, 0, 4)$ Taylor S. Numerade Educator ### Problem 29 In Exercises 27-36, find the distance between the points.$(3, 2, 5), \quad (7, 4, 8)$ Ernest C. Numerade Educator ### Problem 30 In Exercises 27-36, find the distance between the points.$(4, 1, 5), \quad (8, 2, 6)$ Taylor S. Numerade Educator ### Problem 31 In Exercises 27-36, find the distance between the points.$(-1, 4, -2), \quad (6, 0, -9)$ Ernest C. Numerade Educator ### Problem 32 In Exercises 27-36, find the distance between the points.$(1, 1, -7), \quad (-2, -3, -7)$ Taylor S. Numerade Educator ### Problem 33 In Exercises 27-36, find the distance between the points.$(0, -3, 0), \quad (1, 0, -10)$ Ernest C. Numerade Educator ### Problem 34 In Exercises 27-36, find the distance between the points.$(2, -4, 0), \quad (0, 6, -3)$ Taylor S. Numerade Educator ### Problem 35 In Exercises 27-36, find the distance between the points.$(6, -9, 1), \quad (-2, -1, 5)$ Ernest C. Numerade Educator ### Problem 36 In Exercises 27-36, find the distance between the points.$(4, 0, -6), \quad (8, 8, 20)$ Taylor S. Numerade Educator ### Problem 37 In Exercises 37-40, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.$(0, 0, 2), \quad (-2, 5, 2), \quad (0, 4, 0)$ Ernest C. Numerade Educator ### Problem 38 In Exercises 37-40, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.$(2, -1, 2), \quad (-4, 4, 1), \quad (-2, 5, 0)$ Taylor S. Numerade Educator ### Problem 39 In Exercises 37-40, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.$(0, 0, 0), \quad (2, 2, 1), \quad (2, -4, 4)$ Ernest C. Numerade Educator ### Problem 40 In Exercises 37-40, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.$(1, 0, 1), \quad (1, 3, 1), \quad (1, 0, 3)$ Taylor S. Numerade Educator ### Problem 41 In Exercises 41-44, 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.$(1, -3, -2), \quad (5, -1, 2), \quad (-1, 1, 2)$ Ernest C. Numerade Educator ### Problem 42 In Exercises 41-44, 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.$(5, 3, 4), \quad (7, 1, 3), \quad (3, 5, 3)$ Taylor S. Numerade Educator ### Problem 43 In Exercises 41-44, 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.$(4, -1, -2), \quad (8, 1, 2), \quad (2, 3, 2)$ Ernest C. Numerade Educator ### Problem 44 In Exercises 41-44, 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.$(1, -2, -1), \quad (3, 0, 0), \quad (3, -6, 3)$ Taylor S. Numerade Educator ### Problem 45 In Exercises 45-52, find the midpoint of the line segment joining the points.$(0, 0, 0), \quad (3, -2, 4)$ Ernest C. Numerade Educator ### Problem 46 In Exercises 45-52, find the midpoint of the line segment joining the points.$(1, 5, -1), \quad (2, 2, 2)$ Taylor S. Numerade Educator ### Problem 47 In Exercises 45-52, find the midpoint of the line segment joining the points.$(3, -6, 10), \quad (-3, 4, 4)$ Ernest C. Numerade Educator ### Problem 48 In Exercises 45-52, find the midpoint of the line segment joining the points.$(-1, 5, -3), \quad (3, 7, -1)$ Taylor S. Numerade Educator ### Problem 49 In Exercises 45-52, find the midpoint of the line segment joining the points.$(-5, -2, 5), \quad (6, 3, -7)$ Ernest C. Numerade Educator ### Problem 50 In Exercises 45-52, find the midpoint of the line segment joining the points.$(0, -2, 5), \quad (4, 2, 7)$ Taylor S. Numerade Educator ### Problem 51 In Exercises 45-52, find the midpoint of the line segment joining the points.$(-2, 8, 10), \quad (7, -4, 2)$ Ernest C. Numerade Educator ### Problem 52 In Exercises 45-52, find the midpoint of the line segment joining the points.$(9, -5, 1), \quad (9, -2, -4)$ Taylor S. Numerade Educator ### Problem 53 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Center:$(3, 2, 4);$radius:$4$ Ernest C. Numerade Educator ### Problem 54 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Center:$(-3, 4, 3);$radius:$2$ Taylor S. Numerade Educator ### Problem 55 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Center:$(5, 0, -2);$radius:$6$ Ernest C. Numerade Educator ### Problem 56 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Center:$(4, -1, 1);$radius:$5$ Taylor S. Numerade Educator ### Problem 57 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Center:$(-3, 7, 5);$radius:$10$ Ernest C. Numerade Educator ### Problem 58 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Center:$(0, 5, -9);$radius:$8$ Taylor S. Numerade Educator ### Problem 59 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter:$(3, 0, 0), (0, 0, 6)$ Ernest C. Numerade Educator ### Problem 60 In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter:$(1, 0, 0), (0, 5, 0)$ Taylor S. Numerade Educator ### Problem 62 In Exercises 61-70, find the center and radius of the sphere.$x^2+y^2+z^2-9x=0$ Taylor S. Numerade Educator ### Problem 63 In Exercises 61-70, find the center and radius of the sphere.$x^2+y^2+z^2-4x+2y-6z+10=0$ Ernest C. Numerade Educator ### Problem 64 In Exercises 61-70, find the center and radius of the sphere.$x^2+y^2+z^2-6x+4y+9=0$ Taylor S. Numerade Educator ### Problem 65 In Exercises 61-70, find the center and radius of the sphere.$x^2+y^2+z^2+4x-8z+19=0$ Ernest C. Numerade Educator ### Problem 66 In Exercises 61-70, find the center and radius of the sphere.$x^2+y^2+z^2-8y-6z+13=0$ Taylor S. Numerade Educator ### Problem 67 In Exercises 61-70, find the center and radius of the sphere.$9x^2+9y^2+9z^2-18x-6y-72z+73=0$ Ernest C. Numerade Educator ### Problem 68 In Exercises 61-70, find the center and radius of the sphere.$2x^2+2y^2+2z^2-2x-6y-4z+5=0$ Taylor S. Numerade Educator ### Problem 69 In Exercises 61-70, find the center and radius of the sphere.$9x^2+9y^2+9z^2-6x+18y+1=0$ Ernest C. Numerade Educator ### Problem 70 In Exercises 61-70, find the center and radius of the sphere.$4x^2+4y^2+4z^2-4x-32y-8z+33=0$ Taylor S. Numerade Educator ### Problem 71 In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.$(x-1)^2+y^2+z^2=36; xz$-trace Ernest C. Numerade Educator ### Problem 72 In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.$x^2+(y+3)^2+z^2=25; yz$-trace Taylor S. Numerade Educator ### Problem 73 In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.$(x+2)^2+(y-3)^2+z^2=9; yz$-trace Ernest C. Numerade Educator ### Problem 74 In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.$x^2+(y-1)^2+(z+1)^2=4; xy$-trace Taylor S. Numerade Educator ### Problem 75 In Exercises 75 and 76, use a three-dimensional graphing utility to graph the sphere.$x^2+y^2+z^2-6x-8y-10z+46=0$ Ernest C. Numerade Educator ### Problem 76 In Exercises 75 and 76, use a three-dimensional graphing utility to graph the sphere.$x^2+y^2+z^2+6y-8z+21=0$ Taylor S. Numerade Educator ### Problem 77 ARCHITECTURE A spherical building has a diameter of 205 feet. The center of the building is placed at the origin of a three-dimensional coordinate system. What is the equation of the sphere? Ernest C. Numerade Educator ### Problem 78 GEOGRAPHY Assume that Earth is a sphere with a radius of 4000 miles. The center of Earth is placed at the origin of a three-dimensional coordinate system. (a) What is the equation of the sphere? (b) Lines of longitude that run north-south could be represented by what trace(s)? What shape would each of these traces form? (c) Lines of latitude that run east-west could be represented by what trace(s)? What shape would each of these traces form? Taylor S. Numerade Educator ### Problem 79 TRUE OR FALSE? In Exercises 79 and 80, determine whether the statement is true or false. Justify your answer. In the ordered triple$(x, y, z)$that represents point$P$in space,$x$is the directed distance from the$xy$-plane to$P$. Ernest C. Numerade Educator ### Problem 80 TRUE OR FALSE? In Exercises 79 and 80, determine whether the statement is true or false. Justify your answer. The surface consisting of all points$(x, y, z)$in space that are the same distance$r$from the point$(h, j, k)$has a circle as its$xy$-trace. Taylor S. Numerade Educator ### Problem 81 THINK ABOUT IT What is the$z$-coordinate of any point in the$xy$-plane? What is the$y$-coordinate of any point in the$xz$-plane? What is the$x$-coordinate of any point in the$yz$-plane? Ernest C. Numerade Educator ### Problem 82 CAPSTONE Find the equation of the sphere that has the points$(3, -2, 6)$and$(-1, 4, 2)$as endpoints of a diameter. Explain how this problem gives you a chance to use these formulas: the Distance Formula in Space, the Midpoint Formula in Space, and the standard equation of a sphere. Taylor S. Numerade Educator ### Problem 83 A sphere intersects the$yz$-plane. Describe the trace. Ernest C. Numerade Educator ### Problem 84 A plane intersects the$xy$-plane. Describe the trace. Taylor S. Numerade Educator ### Problem 85 A line segment has$(x_1, y_1, z_1)$as one endpoint and$(x_m, y_m, z_m)$as its midpoint. Find the other endpoint$(x_2, y_2, z_2)$of the line segment in terms of$x_1, y_1, z_1, x_m, y_m,$and$z_m$. Ernest C. Numerade Educator ### Problem 86 Use the result of Exercise 85 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and the midpoint are$(3, 0, 2)$and$(5, 8, 7),\$ respectively. Taylor S.