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Thomas Calculus

George B. Thomas, Jr.

Chapter 12

Vectors and the Geometry of Space

Educators


Problem 1

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x=2, \quad y=3$$

Elizabeth X.
Numerade Educator

Problem 2

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x=-1, \quad z=0$$

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Problem 3

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$y=0, \quad z=0$$

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Problem 4

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x=1, \quad y=0$$

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Problem 5

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+y^{2}=4, \quad z=0$$

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Problem 6

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+y^{2}=4, \quad z=-2$$

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Problem 7

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+z^{2}=4, \quad y=0$$

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Problem 8

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$y^{2}+z^{2}=1, \quad x=0$$

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Problem 9

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+y^{2}+z^{2}=1, \quad x=0$$

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Problem 10

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+y^{2}+z^{2}=25, \quad y=-4$$

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Problem 11

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+y^{2}+(z+3)^{2}=25, \quad z=0$$

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Problem 12

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+(y-1)^{2}+z^{2}=4, \quad y=0$$

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Problem 13

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+y^{2}=4, \quad z=y$$

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Problem 14

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$x^{2}+y^{2}+z^{2}=4, \quad y=x$$

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Problem 15

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$y=x^{2}, \quad z=0$$

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Problem 16

In Exercises $1-16,$ give a geometric description of the set of points in
space whose coordinates satisfy the given pairs of equations.
$$z=y^{2}, \quad x=1$$

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Problem 17

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\text { b. } x \geq 0, \quad y \geq 0, \quad z=0 \quad \text { b. } x \geq 0, \quad y \leq 0, \quad z=0
$$

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Problem 18

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\begin{array}{ll}{\text { a. } 0 \leq x \leq 1} & {\text { b. } 0 \leq x \leq 1, \quad 0 \leq y \leq 1} \\ {\text { c. } 0 \leq x \leq 1,} & {0 \leq y \leq 1, \quad 0 \leq z \leq 1}\end{array}
$$

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Problem 19

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\text { a. } x^{2}+y^{2}+z^{2} \leq 1 \quad \text { b. } x^{2}+y^{2}+z^{2}>1
$$

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Problem 20

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\begin{array}{l}{\text { a. } x^{2}+y^{2} \leq 1, \quad z=0 \quad \text { b. } x^{2}+y^{2} \leq 1, \quad z=3} \\ {\text { c. } x^{2}+y^{2} \leq 1, \text { no restriction on } z}\end{array}
$$

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Problem 21

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\begin{array}{l}{\text { a. } 1 \leq x^{2}+y^{2}+z^{2} \leq 4} \\ {\text { b. } x^{2}+y^{2}+z^{2} \leq 1, \quad z \geq 0}\end{array}
$$

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Problem 22

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\text { a. } x=y, \quad z=0 \quad \text { b. } x=y, \text { no restriction on } z
$$

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Problem 23

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\text { a. } y \geq x^{2}, \quad z \geq 0 \quad \text { b. } x \leq y^{2}, \quad 0 \leq z \leq 2
$$

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Problem 24

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and
inequalities.
$$
\begin{array}{l}{\text { a. } z=1-y, \quad \text { no restriction on } x} \\ {\text { b. } z=y^{3}, \quad x=2}\end{array}
$$

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Problem 25

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The plane perpendicular to the
$$\begin{array}{ll}{\text { a. } x \text { -axis at }(3,0,0)} & {\text { b. } y \text { -axis at }(0,-1,0)} \\ {\text { c. }} & {z \text { -axis at }(0,0,-2)}\end{array}$$

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Problem 26

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The plane through the point $(3,-1,2)$ perpendicular to the
$$\begin{array}{llll}{\text { a. } x \text { -axis }} & {\text { b. } y \text { -axis }} & {\text { c. } z \text { -axis }}\end{array}$$

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Problem 28

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The circle of radius 2 centered at $(0,0,0)$ and lying in the
$$
\text { a. } x y \text { -plane } \quad \text { b. } y z \text { -plane } \quad \text { c. } x z
$$

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Problem 29

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The circle of radius 2 centered at $(0,2,0)$ and lying in the
$$
\text { a. } x \text { y-plane } \quad \text { b. } y z \text { -plane } \quad \text { c. plane } y=2
$$

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Problem 30

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The circle of radius 1 centered at $(-3,4,1)$ and lying in a plane
parallel to the
$$
\text { a. } x y \text { -plane } \quad \text { b. } y z \text { -plane } \quad \text { c. } x z
$$

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Problem 31

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The line through the point $(1,3,-1)$ parallel to the
$$
\text { a. } x \text { -axis } \quad \text { b. } y \text { -axis } \quad \text { c. } z
$$

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Problem 32

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The set of points in space equidistant from the origin and the
point $(0,2,0)$

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Problem 33

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The circle in which the plane through the point $(1,1,3)$ perpendicular to the $z$ -axis meets the sphere of radius 5 centered at the
origin

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Problem 34

In Exercises $25-34$ , describe the given set with a single equation or
with a pair of equations.
The set of points in space that lie 2 units from the point $(0,0,1)$
and, at the same time, 2 units from the point $(0,0,-1)$

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Problem 35

Write inequalities to describe the sets in Exercises $35-40$
The slab bounded by the planes $z=0$ and $z=1$ (planes
included)

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Problem 36

Write inequalities to describe the sets in Exercises $35-40$
The solid cube in the first octant bounded by the coordinate
planes and the planes $x=2, y=2,$ and $z=2$

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Problem 37

Write inequalities to describe the sets in Exercises $35-40$
The half-space consisting of the points on and below the $x y$ -plane

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Problem 38

Write inequalities to describe the sets in Exercises $35-40$
The upper hemisphere of the sphere of radius 1 centered at the
origin

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Problem 39

Write inequalities to describe the sets in Exercises $35-40$
The (a) interior and (b) exterior of the sphere of radius 1 centered
at the point $(1,1,1)$

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Problem 41

In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}$
$$P_{1}(1,1,1), \quad P_{2}(3,3,0)$$

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Problem 42

In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}$
$$P_{1}(-1,1,5), \quad P_{2}(2,5,0)$$

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Problem 43

In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}$
$$P_{1}(1,4,5), \quad P_{2}(4,-2,7)$$

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Problem 44

In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}$
$$P_{1}(3,4,5), \quad P_{2}(2,3,4)$$

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Problem 45

In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}$
$$P_{1}(0,0,0), \quad P_{2}(2,-2,-2)$$

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Problem 46

In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}$
$$P_{1}(5,3,-2), \quad P_{2}(0,0,0)$$

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Problem 47

Find the centers and radii of the spheres in Exercises $47-50$
$$(x+2)^{2}+y^{2}+(z-2)^{2}=8$$

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Problem 48

Find the centers and radii of the spheres in Exercises $47-50$
$$(x-1)^{2}+\left(y+\frac{1}{2}\right)^{2}+(z+3)^{2}=25$$

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Problem 49

Find the centers and radii of the spheres in Exercises $47-50$
$$(x-\sqrt{2})^{2}+(y-\sqrt{2})^{2}+(z+\sqrt{2})^{2}=2$$

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Problem 50

Find the centers and radii of the spheres in Exercises $47-50$
$$x^{2}+\left(y+\frac{1}{3}\right)^{2}+\left(z-\frac{1}{3}\right)^{2}=\frac{16}{9}$$

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Problem 51

Find equations for the spheres whose centers and radii are given in
Exercises $51-54 .$
$\frac{\text { Center }}{(1,2,3)} \frac{\text { Radius }}{\sqrt{14}}$

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Problem 52

Find equations for the spheres whose centers and radii are given in
Exercises $51-54 .$
$$\frac{\text { Center }}{(0,-1,5)} \frac{\text { Radius }}{2}$$

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Problem 53

Find equations for the spheres whose centers and radii are given in
Exercises $51-54 .$
$$\frac{\text { Center }}{\left(-1, \frac{1}{2},-\frac{2}{3}\right)} \frac{\text { Radius }}{\frac{4}{9}}$$

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Problem 54

Find equations for the spheres whose centers and radii are given in
Exercises $51-54 .$
$$\frac{\text { Center }}{(0,-7,0)} \frac{\text { Radius }}{7}$$

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Problem 55

Find the centers and radii of the spheres in Exercises $55-58 .$
$$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

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Problem 56

Find the centers and radii of the spheres in Exercises $55-58 .$
$$x^{2}+y^{2}+z^{2}-6 y+8 z=0$$

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Problem 57

Find the centers and radii of the spheres in Exercises $55-58 .$
$$2 x^{2}+2 y^{2}+2 z^{2}+x+y+z=9$$

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Problem 58

Find the centers and radii of the spheres in Exercises $55-58 .$
$$3 x^{2}+3 y^{2}+3 z^{2}+2 y-2 z=9$$

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Problem 59

Find a formula for the distance from the point $P(x, y, z)$ to the
$$
\text { a. } x \text { -axis. } \quad \text { b. } y \text { -axis. } \quad \text { c. } z
$$

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Problem 60

Find a formula for the distance from the point $P(x, y, z)$ to the
$$
\text { a. } x y \text { -plane. } \quad \text { b. } y z \text { -plane. } \quad \text { c. } x z
$$

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Problem 61

Find the perimeter of the triangle with vertices $A(-1,2,1),$
$B(1,-1,3),$ and $C(3,4,5) .$

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Problem 62

Show that the point $P(3,1,2)$ is equidistant from the points
$A(2,-1,3)$ and $B(4,3,1)$ .

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Problem 63

Find an equation for the set of all points equidistant from the
planes $y=3$ and $y=-1 .$

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Problem 64

Find an equation for the set of all points equidistant from the
point $(0,0,2)$ and the $x y$ -plane.

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Problem 65

Find the point on the sphere $x^{2}+(y-3)^{2}+(z+5)^{2}=4$
nearest
$$
\text{a.} x y \text { -plane. } \quad \text { b. the point }(0,7,-5)
$$

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Problem 66

Find the point equidistant from the points $(0,0,0),(0,4,0),$
$(3,0,0),$ and $(2,2,-3) .$

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