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Precalculus with Limits (2010)

Ron Larson, David C. Falvo

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.

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Problem 2

The three coordinate planes of a three-dimensional coordinate system are the _______ , the _______ ,
and the _______ .

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Problem 3

The coordinate planes of a three-dimensional coordinate system separate the coordinate system into eight _______ .

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Problem 4

The distance between the points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ can be found using the _____ _____ in Space.

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Problem 5

The midpoint of the line segment joining the points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ given by the Midpoint Formula in Space is _____.

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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 $r$.

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Problem 7

A _____ in _____ is the collection of points satisfying an equation involving $x, y,$ and $z$.

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Problem 8

The intersection of a surface with one of the three coordinate planes is called a _______ of the surface.

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Problem 9

In Exercises 9 and $10,$ approximate the coordinates of the points.

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Problem 10

In Exercises 9 and $10,$ approximate the coordinates of the points.

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Problem 11

In Exercises $11-16,$ plot each point in the same three-dimensional coordinate system.
$$
\begin{array}{l}{\text { (a) }(2,1,3)} \\ {\text { (b) }(1,-1,-2)}\end{array}
$$

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Problem 12

In Exercises $11-16,$ plot each point in the same three-dimensional coordinate system.
$$
\begin{array}{l}{\text { (a) }(3,0,0)} \\ {\text { (b) }(-3,-2,-1)}\end{array}
$$

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Problem 13

In Exercises $11-16,$ plot each point in the same three-dimensional coordinate system.
$$
\begin{array}{l}{\text { (a) }(3,-1,0)} \\ {\text { (b) }(-4,2,2)}\end{array}
$$

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Problem 14

In Exercises $11-16,$ plot each point in the same three-dimensional coordinate system.
$$
\begin{array}{l}{\text { (a) }(0,4,-3)} \\ {\text { (b) }(4,0,4)}\end{array}
$$

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Problem 15

In Exercises $11-16,$ plot each point in the same three-dimensional coordinate system.
$$
\begin{array}{l}{\text { (a) }(3,-2,5)} \\ {\text { (b) }\left(\frac{3}{2}, 4,-2\right)}\end{array}
$$

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Problem 16

In Exercises $11-16,$ plot each point in the same three-dimensional coordinate system.
$$
\begin{array}{l}{\text { (a) }(5,-2,2)} \\ {\text { (b) }(5,-2,-2)}\end{array}
$$

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Problem 17

In Exercises $17-20,$ find the coordinates of the point.
The point is located three units behind the $y z$ -plane, four units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

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Problem 18

In Exercises $17-20,$ find the coordinates of the point.
The point is located seven units in front of the $y z$ -plane, two units to the left of the $x z$ -plane, and one unit below the $x y$ -plane.

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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.

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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 $x z$ -plane, and six units above the $x y$ -plane.

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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.
$$
x > 0, y < 0, z > 0
$$

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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.
$$
x < 0, y > 0, z < 0
$$

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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.
$$
z > 0
$$

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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.
$$
y < 0
$$

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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.
$$
x y < 0
$$

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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.
$$
y z > 0
$$

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Problem 27

In Exercises $27-36,$ find the distance between the points.
$$
(0,0,0),(5,2,6)
$$

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Problem 28

In Exercises $27-36,$ find the distance between the points.
$$
(1,0,0),(7,0,4)
$$

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Problem 29

In Exercises $27-36,$ find the distance between the points.
$$
(3,2,5),(7,4,8)
$$

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Problem 30

In Exercises $27-36,$ find the distance between the points.
$$
(4,1,5),(8,2,6)
$$

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Problem 31

In Exercises $27-36,$ find the distance between the points.
$$
(-1,4,-2),(6,0,-9)
$$

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Problem 32

In Exercises $27-36,$ find the distance between the points.
$$
(1,1,-7),(-2,-3,-7)
$$

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Problem 33

In Exercises $27-36,$ find the distance between the points.
$$
(0,-3,0),(1,0,-10)
$$

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Problem 34

In Exercises $27-36,$ find the distance between the points.
$$
(2,-4,0),(0,6,-3)
$$

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Problem 35

In Exercises $27-36,$ find the distance between the points.
$$
(6,-9,1),(-2,-1,5)
$$

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Problem 36

In Exercises $27-36,$ find the distance between the points.
$$
(4,0,-6),(8,8,20)
$$

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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.
$$
(0,0,2),(-2,5,2),(0,4,0)
$$

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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.
$$
(2,-1,2),(-4,4,1),(-2,5,0)
$$

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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.
$$
(0,0,0),(2,2,1),(2,-4,4)
$$

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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.
$$
(1,0,1),(1,3,1),(1,0,3)
$$

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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.
$$
(1,-3,-2),(5,-1,2),(-1,1,2)
$$

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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.
$$
(5,3,4),(7,1,3),(3,5,3)
$$

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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.
$$
(4,-1,-2),(8,1,2),(2,3,2)
$$

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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.
$$
(1,-2,-1),(3,0,0),(3,-6,3)
$$

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Problem 45

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(0,0,0),(3,-2,4)
$$

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Problem 46

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(1,5,-1),(2,2,2)
$$

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Problem 47

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(3,-6,10),(-3,4,4)
$$

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Problem 48

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(-1,5,-3),(3,7,-1)
$$

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Problem 49

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(-5,-2,5),(6,3,-7)
$$

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Problem 50

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(0,-2,5),(4,2,7)
$$

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Problem 51

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(-2,8,10),(7,-4,2)
$$

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Problem 52

In Exercises $45-52,$ find the midpoint of the line segment joining the points.
$$
(9,-5,1),(9,-2,-4)
$$

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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

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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

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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

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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

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Problem 57

In Exercises $53-60,$ find the standard form of the equation of the sphere with the given characteristics.
Center: $(-3,7,5) ;$ diameter: 10

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Problem 58

In Exercises $53-60,$ find the standard form of the equation of the sphere with the given characteristics.
Center: $(0,5,-9) ;$ diameter: 8

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Problem 59

In Exercises $53-60,$ find the standard form of the equation of the sphere with the given characteristics.
Endpoints of a diameter: $(3,0,0),(0,0,6)$

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Problem 60

In Exercises $53-60,$ find the standard form of the equation of the sphere with the given characteristics.
Endpoints of a diameter: $(1,0,0),(0,5,0)$

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Problem 61

In Exercises $61-70,$ find the center and radius of the sphere.
$$
x^{2}+y^{2}+z^{2}-6 x=0
$$

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Problem 62

In Exercises $61-70,$ find the center and radius of the sphere.
$$
x^{2}+y^{2}+z^{2}-9 x=0
$$

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Problem 63

In Exercises $61-70,$ find the center and radius of the sphere.
$$
x^{2}+y^{2}+z^{2}-4 x+2 y-6 z+10=0
$$

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Problem 64

In Exercises $61-70,$ find the center and radius of the sphere.
$$
x^{2}+y^{2}+z^{2}-6 x+4 y+9=0
$$

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Problem 65

In Exercises $61-70,$ find the center and radius of the sphere.
$$
x^{2}+y^{2}+z^{2}+4 x-8 z+19=0
$$

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Problem 66

In Exercises $61-70,$ find the center and radius of the sphere.
$$
x^{2}+y^{2}+z^{2}-8 y-6 z+13=0
$$

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Problem 67

In Exercises $61-70,$ find the center and radius of the sphere.
$$
9 x^{2}+9 y^{2}+9 z^{2}-18 x-6 y-72 z+73=0
$$

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Problem 68

In Exercises $61-70,$ find the center and radius of the sphere.
$$
2 x^{2}+2 y^{2}+2 z^{2}-2 x-6 y-4 z+5=0
$$

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Problem 69

In Exercises $61-70,$ find the center and radius of the sphere.
$$
9 x^{2}+9 y^{2}+9 z^{2}-6 x+18 y+1=0
$$

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Problem 70

In Exercises $61-70,$ find the center and radius of the sphere.
$$
4 x^{2}+4 y^{2}+4 z^{2}-4 x-32 y+8 z+33=0
$$

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Problem 71

In Exercises $71-74$ , sketch the graph of the equation and sketch the specified trace.
$(x-1)^{2}+y^{2}+z^{2}=36 ; x z$ -trace

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Problem 72

In Exercises $71-74$ , sketch the graph of the equation and sketch the specified trace.
$x^{2}+(y+3)^{2}+z^{2}=25 ; y z$ -trace

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Problem 73

In Exercises $71-74$ , sketch the graph of the equation and sketch the specified trace.
$(x+2)^{2}+(y-3)^{2}+z^{2}=9 ; \quad y z$ -trace

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Problem 74

In Exercises $71-74$ , sketch the graph of the equation and sketch the specified trace.
$x^{2}+(y-1)^{2}+(z+1)^{2}=4 ; x y$ -trace

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Problem 75

In Exercises 75 and 76 , use a three-dimensional graphing utility to graph the sphere.
$$
x^{2}+y^{2}+z^{2}-6 x-8 y-10 z+46=0
$$

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Problem 76

In Exercises 75 and 76 , use a three-dimensional graphing utility to graph the sphere.
$$
x^{2}+y^{2}+z^{2}+6 y-8 z+21=0
$$

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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?

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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?

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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 $x y$ -plane to $P .$

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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 $x y$ -trace.

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Problem 81

THINK ABOUT IT What is the z-coordinate of any point in the $x y$ -plane? What is the $y$ -coordinate of any point in the $x z$ -plane? What is the $x$ -coordinate of any point in the $y z$ -plane?

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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.

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Problem 83

A sphere intersects the $y z$ -plane. Describe the trace.

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Problem 84

A plane intersects the $x y$ -plane. Describe the trace.

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Problem 85

A line segment has $\left(x_{1}, y_{1}, z_{1}\right)$ as one endpoint and $\left(x_{m}, y_{m}, z_{m}\right)$ as its midpoint. Find the other endpoint $\left(x_{2}, y_{2}, z_{2}\right)$ of the line segment in terms of $x_{1}, y_{1}$ $z_{1}, x_{m}, y_{m},$ and $z_{m}$ .

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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.

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