University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 11

Vectors and the Geometry of Space

Educators


Problem 1

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.
$$x=2, y=3$$

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Problem 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.
$$\text { a. } x \geq 0, \quad y \geq 0, \quad z=0 \quad \text { b. } x \geq 0, \quad y \leq 0, \quad z=0$$

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Problem 18

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.
a. $0 \leq x \leq 1$
b. $0 \leq x \leq 1, \quad 0 \leq y \leq 1$
c. $0 \leq x \leq 1, \quad 0 \leq y \leq 1, \quad 0 \leq z \leq 1$

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Problem 19

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.
a. $x^{2}+y^{2}+z^{2} \leq 1$
b. $x^{2}+y^{2}+z^{2}>1$

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Problem 20

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.
a. $x^{2}+y^{2} \leq 1, \quad z=0$
b. $x^{2}+y^{2} \leq 1, \quad z=3$
c. $x^{2}+y^{2} \leq 1,$ no restriction on $z$

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Problem 21

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.
a. $1 \leq x^{2}+y^{2}+z^{2} \leq 4$
b. $x^{2}+y^{2}+z^{2} \leq 1, \quad z \geq 0$

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Problem 22

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.
a. $x=y, z=0$
b. $x=y, \quad$ no restriction on $z$

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Problem 23

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

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Problem 24

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.
a. $z=1-y, \quad$ no restriction on $x$
b. $z=y^{3}, \quad x=2$

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Problem 25

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 26

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 27

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 28

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 29

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 30

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 31

Find the distance from the point (3,-4,2) to the
a. $x y$ -plane
b. yz-plane
c. $x z$ -plane

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Problem 32

Find the distance from the point (-2,1,4) to the
a. plane $x=3$
b. plane $y=-5$
c. plane $z=-1$

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Problem 33

Find the distance from the point (4,3,0) to the
a. $x$ -axis
b. $y$ -axis
c. z-axis

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Problem 34

Find the distance from the
a. $x$ -axis to the plane $z=3$
b. origin to the plane $2=z-x$
c. point (0,4,0) to the plane $y=x$

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Problem 35

Describe the given set with a single equation or with a pair of equations.
The plane perpendicular to the
a. $x$ -axis at (3,0,0)
b. $y$ -axis at (0,-1,0)
c. $z$ -axis at (0,0,-2)

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Problem 36

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
a. $x$ -axis
b. y-axis
c. z-axis

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Problem 37

Describe the given set with a single equation or with a pair of equations.
The plane through the point (3,-1,1) parallel to the
a. $x y$ -plane
b. yz-plane
c. $x z$ -plane

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Problem 38

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
a. $x y$ -plane
b. $y z$ -plane
c. $x z$ -plane

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Problem 39

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
a. $x y$ -plane
b. $y z$ -plane
c. plane $y=2$

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Problem 40

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
a. $x y$ -plane
b. $y z$ -plane
c. $x z$ -plane

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Problem 41

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
a. $x$ -axis
b. $y$ -axis
c. $z$ -axis

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Problem 42

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 43

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 44

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 45

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

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Problem 46

Write inequalities to describe the sets.
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 47

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

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Problem 48

Write inequalities to describe the sets.
The upper hemisphere of the sphere of radius 1 centered at the origin

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Problem 49

Write inequalities to describe the sets.
The (a) interior and (b) exterior of the sphere of radius 1 centered at the point (1,1,1)

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Problem 50

Write inequalities to describe the sets.
The closed region bounded by the spheres of radius 1 and radius 2 centered at the origin. (Closed means the spheres are to be included. Had we wanted the spheres left out, we would have asked for the open region bounded by the spheres. This is analogous to the way we use closed and open to describe intervals: closed means endpoints included, open means endpoints left out. Closed sets include boundaries; open sets leave them out.)

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Problem 51

Find the center $C$ and the radius $a$ for the spheres.
$$(x+2)^{2}+y^{2}+(z-2)^{2}=8$$

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Problem 52

Find the center $C$ and the radius $a$ for the spheres.
$$(x-1)^{2}+\left(y+\frac{1}{2}\right)^{2}+(z+3)^{2}=25$$

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Problem 53

Find the center $C$ and the radius $a$ for the spheres.
$$(x-\sqrt{2})^{2}+(y-\sqrt{2})^{2}+(z+\sqrt{2})^{2}=2$$

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Problem 54

Find the center $C$ and the radius $a$ for the spheres.
$$x^{2}+\left(y+\frac{1}{3}\right)^{2}+\left(z-\frac{1}{3}\right)^{2}=\frac{16}{9}$$

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Problem 55

Find the center $C$ and the radius $a$ for the spheres.
$$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

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Problem 56

Find the center $C$ and the radius $a$ for the spheres.
$$x^{2}+y^{2}+z^{2}-6 y+8 z=0$$

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Problem 57

Find the center $C$ and the radius $a$ for the spheres.
$$2 x^{2}+2 y^{2}+2 z^{2}+x+y+z=9$$

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Problem 58

Find the center $C$ and the radius $a$ for the spheres.
$$3 x^{2}+3 y^{2}+3 z^{2}+2 y-2 z=9$$

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Problem 59

Find the center $C$ and the radius $a$ for the spheres.
$$x^{2}+y^{2}+z^{2}-4 x+6 y-10 z=11$$

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Problem 60

Find the center $C$ and the radius $a$ for the spheres.
$$(x-1)^{2}+(y-2)^{2}+(z+1)^{2}=103+2 x+4 y-2 z$$

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Problem 61

Find equations for the spheres whose centers and radii are given.
$$\begin{array}{l}\text { (Radius) } \\\sqrt{14} \\2 \\\frac{4}{9} \\7\end{array}$$
(Center)
$$(1,2,3)$$

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Problem 62

Find equations for the spheres whose centers and radii are given.
$$\begin{array}{l}\text { (Radius) } \\\sqrt{14} \\2 \\\frac{4}{9} \\7\end{array}$$
(Center)
$$(0,-1,5)$$

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Problem 63

Find equations for the spheres whose centers and radii are given.
$$\begin{array}{l}\text { (Radius) } \\\sqrt{14} \\2 \\\frac{4}{9} \\7\end{array}$$
(Center)
$$\left(-1, \frac{1}{2},-\frac{2}{3}\right)$$

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Problem 64

Find equations for the spheres whose centers and radii are given.
$$\begin{array}{l}\text { (Radius) } \\\sqrt{14} \\2 \\\frac{4}{9} \\7\end{array}$$
(Center)
$$(0,-7,0)$$

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Problem 65

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

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Problem 66

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

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Problem 67

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 68

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 69

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

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Problem 70

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 71

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

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Problem 72

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

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Problem 73

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

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Problem 74

Find an equation for the set of points equidistant from the $y$ -axis and the plane $z=6$

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Problem 75

Find an equation for the set of points equidistant from the
a. $x y$ -plane and the $y z$ -plane.
b. $x$ -axis and the $y$ -axis.

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Problem 76

$.$ Find all points that simultaneously lie 3 units from each of the points $(2,0,0),(0,2,0),$ and (0,0,2)

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