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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.
Numerade Educator