Chapter 11

Analytic Geometry in Three Dimensions

Educators

NH
SZ

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 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 $r$.

NH
Nashelia H.
Numerade Educator

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

Plotting Points in Space In Exercises $9-14,$ plot both
points 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 10

Plotting Points in Space In Exercises $9-14,$ plot both
points 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 11

Plotting Points in Space In Exercises $9-14,$ plot both
points 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 12

Plotting Points in Space In Exercises $9-14,$ plot both
points 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 13

Plotting Points in Space In Exercises $9-14,$ plot both
points 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 14

Plotting Points in Space In Exercises $9-14,$ plot both
points 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 15

Finding the Coordinates of a Point In Exercises
$15-20$ , 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 16

Plotting Points in Space In Exercises $9-14,$ plot both
points in the same three-dimensional coordinate system.
The point is located seven units in front of the yz-plane,
two units to the left of the $x$ -plane, and one unit below
the $x y$ -plane.

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Problem 17

Plotting Points in Space In Exercises $9-14,$ plot both
points in the same three-dimensional coordinate system.
The point is located on the $x$ -axis, eight units in front of
the $y z$ -plane.

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Problem 18

Plotting Points in Space In Exercises $9-14,$ plot both
points in the same three-dimensional coordinate system.
The point is located on the $y$ -axis, 11 units in front of
the $x$ -plane.

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Problem 19

Plotting Points in Space In Exercises $9-14,$ plot both
points in the same three-dimensional coordinate system.
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 20

Plotting Points in Space In Exercises $9-14,$ plot both
points in the same three-dimensional coordinate system.
The point is located in the $x$ -plane, two units to the right
of the yz-plane, and seven units above the $x y$ -plane.

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Problem 21

Determining Octants 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

Determining Octants 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

Determining Octants 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

Determining Octants 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

Determining Octants 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

Determining Octants 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

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(0,0,0),(5,2,6)$$

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Problem 28

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(1,0,0),(7,0,4)$$

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Problem 29

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(3,2,5),(7,4,8)$$

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Problem 30

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(4,1,5),(8,2,6)$$

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Problem 31

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(-1,4,-2),(6,0,-9)$$

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Problem 32

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(1,1,-7),(-2,-3,-7)$$

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Problem 33

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(0,-3,0),(1,0,-10)$$

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Problem 34

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(2,-4,0),(0,6,-3)$$

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Problem 35

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(6,-9,1),(-2,-1,5)$$

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Problem 36

Finding the Distance Between Two Points in Space
In Exercises $27-36,$ tind the distance between the points.
$$(4,0,-6),(8,8,20)$$

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Problem 37

Using the Pythagorean Theorem 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.

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Problem 38

Using the Pythagorean Theorem 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.

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Problem 39

Using the Pythagorean Theorem 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

Using the Pythagorean Theorem 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

Finding the Side Lengths of a Triangle 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

Finding the Side Lengths of a Triangle 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

Finding the Side Lengths of a Triangle 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

Finding the Side Lengths of a Triangle 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

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

Jennifer D.
Numerade Educator

Problem 46

Using the Midpoint Formula in Space 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

Using the Midpoint Formula in Space 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

Using the Midpoint Formula in Space 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 50

Using the Midpoint Formula in Space 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

Using the Midpoint Formula in Space 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

Using the Midpoint Formula in Space 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

Finding the Equation of a Sphere In Exercises
$53-60$ , find the standard form of the equation of the
sphere with the given characteristics.
Center: $$(3,2,4) ; \text { radius: } 4$$

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Problem 54

Using the Midpoint Formula in Space In
Exercises $45-52$ , find the midpoint of the line segment
joining the points.
Center:$$(-3,4,3) ; \text { radius: } 2$$

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Problem 55

Using the Midpoint Formula in Space In
Exercises $45-52$ , find the midpoint of the line segment
joining the points.
Center:$$(5,0,-2) ; \text { radius: } 6$$

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Problem 56

Using the Midpoint Formula in Space In
Exercises $45-52$ , find the midpoint of the line segment
joining the points.
Center:$$(4,-1,1) ; \text { radius: } 5$$

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Problem 57

Using the Midpoint Formula in Space In
Exercises $45-52$ , find the midpoint of the line segment
joining the points.
Center:$$(-3,7,5) ; \text { diameter: } 10$$

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Problem 58

Using the Midpoint Formula in Space In
Exercises $45-52$ , find the midpoint of the line segment
joining the points.
Center:$$(0,5,-9) ; \text { diameter: } 8$$

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Problem 59

Using the Midpoint Formula in Space In
Exercises $45-52$ , find the midpoint of the line segment
joining the points.
Endpoints of a diameter $$(3,0,0),(0,0,6)$$

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Problem 60

Using the Midpoint Formula in Space In
Exercises $45-52$ , find the midpoint of the line segment
joining the points.
Endpoint of a diameter:$$(1,0,0),(0,5,0)$$

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Problem 61

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

Finding the Center and Radius of a Sphere 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

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

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Problem 72

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

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Problem 73

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

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Problem 74

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

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Problem 75

Graphing a Sphere In Exercises $75-78,$ 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

Graphing a Sphere In Exercises $75-78,$ 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

Graphing a Sphere In Exercises $75-78,$ use a
three-dimensional graphing utility to graph the sphere.
$$4 x^{2}+4 y^{2}+4 z^{2}-8 x-16 y+8 z-25=0$$

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Problem 78

Graphing a Sphere In Exercises $75-78,$ use a
three-dimensional graphing utility to graph the sphere.
$$9 x^{2}+9 y^{2}+9 z^{2}+18 x-18 y+36 z+35=0$$

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Problem 79

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 80

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?

SZ
Scott Z.
Numerade Educator

Problem 81

True or False? In Exercises 81 and $82,$ 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 82

True or False? In Exercises 81 and $82,$ determine
whether the statement is true or false. Justify your answer.
A sphere has a circle as its $x y$ -trace.

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Problem 83

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 yz-plane?

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Problem 84

How DO YOU SEE IT? Approximate the
coordinates of each point, and state the octant
in which each point lies.

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Problem 85

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

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Problem 86

Finding an Endpoint Use the result of Exercise 85
to find the coordinates of the endpoint of a line segment
when the coordinates of the other endpoint and the
midpoint are $(3,0,2)$ and $(5,8,7),$ respectively.

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