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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The three coordinate planes of a three-dimensional coordinate system are the______________ , the ________________ , and the_____________.

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

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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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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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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 $r$.

Nashelia H.

Numerade Educator

A____________in___________is the collection of points satisfying an equation involving $x, y,$ and $z$

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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?

Scott Z.

Numerade Educator

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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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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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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How DO YOU SEE IT? Approximate the

coordinates of each point, and state the octant

in which each point lies.

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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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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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