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Calculus Multivariable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 17

Parameterization and Vector Fields - all with Video Answers

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Section 1

Parameterized Curves

01:44

Problem 1

Find a parameterization for the curve. (FIGURE CAN'T COPY)

WM
William Mead
Numerade Educator
01:01

Problem 2

Find a parameterization for the curve. (FIGURE CAN'T COPY)

WM
William Mead
Numerade Educator
01:26

Problem 3

Find a parameterization for the curve. (FIGURE CAN'T COPY)

WM
William Mead
Numerade Educator
01:38

Problem 4

Find a parameterization for the curve. (FIGURE CAN'T COPY)

WM
William Mead
Numerade Educator
00:50

Problem 5

Find a parameterization for the curve. (FIGURE CAN'T COPY)

WM
William Mead
Numerade Educator
02:38

Problem 6

Find a parameterization for the curve. (FIGURE CAN'T COPY)

WM
William Mead
Numerade Educator
01:41

Problem 7

Find parametric equations for the line.
The line in the direction of the vector $\vec{i}-\vec{k}$ and through the point (0,1,0).

Khushbu Rani
Khushbu Rani
Numerade Educator
01:41

Problem 8

Find parametric equations for the line.
The line in the direction of the vector $\vec{i}+2 \vec{j}-\vec{k}$ and through the point (3,0,-4).

Khushbu Rani
Khushbu Rani
Numerade Educator
01:38

Problem 9

Find parametric equations for the line.
The line parallel to the $z$ -axis passing through the point (1,0,0).

Khushbu Rani
Khushbu Rani
Numerade Educator
02:05

Problem 10

Find parametric equations for the line.
The line in the direction of the vector $5 \vec{j}+2 \vec{k}$ and through the point (5,-1,1).

Khushbu Rani
Khushbu Rani
Numerade Educator
01:39

Problem 11

Find parametric equations for the line.
The line in the direction of the vector $3 \vec{i}-3 \vec{j}+\vec{k}$ and through the point (1,2,3).

Khushbu Rani
Khushbu Rani
Numerade Educator
02:03

Problem 12

Find parametric equations for the line.
The line in the direction of the vector $2 \vec{i}+2 \vec{j}-3 \vec{k}$ and through the point (-3,4,-2).

Khushbu Rani
Khushbu Rani
Numerade Educator
01:50

Problem 13

Find parametric equations for the line.
The line through (-3,-2,1) and (-1,-3,-1).

Khushbu Rani
Khushbu Rani
Numerade Educator
01:55

Problem 14

Find parametric equations for the line.
The line through the points (1,5,2) and (5,0,-1).

Khushbu Rani
Khushbu Rani
Numerade Educator
01:47

Problem 15

Find parametric equations for the line.
The line through the points (2,3,-1) and (5,2,0).

Khushbu Rani
Khushbu Rani
Numerade Educator
01:50

Problem 16

Find parametric equations for the line.
The line through (3,-2,2) and intersecting the $y$ -axis at $y=2$.

Khushbu Rani
Khushbu Rani
Numerade Educator
02:10

Problem 17

Find parametric equations for the line.
The line intersecting the $x$ -axis at $x=3$ and the $z$ -axis at $z=-5$.

Khushbu Rani
Khushbu Rani
Numerade Educator
01:57

Problem 18

Find a parameterization for the curve.
A line segment between (2,1,3) and (4,3,2).

Khushbu Rani
Khushbu Rani
Numerade Educator
02:13

Problem 19

Find a parameterization for the curve.
A circle of radius 3 centered on the $z$ -axis and lying in the plane $z=5$.

William Semus
William Semus
Numerade Educator
02:04

Problem 20

Find a parameterization for the curve.
A line perpendicular to the plane $z=2 x-3 y+7$ and through the point (1,1,6).

Khushbu Rani
Khushbu Rani
Numerade Educator
View

Problem 21

Find a parameterization for the curve.
The circle of radius 2 in the $x y$ -plane, centered at the origin, clockwise.

Bradley Duda
Bradley Duda
Numerade Educator
View

Problem 22

Find a parameterization for the curve.
The circle of radius 2 parallel to the $x y$ -plane, centered at the point $(0,0,1),$ and traversed counterclockwise when viewed from below.

Bradley Duda
Bradley Duda
Numerade Educator
View

Problem 23

Find a parameterization for the curve.
The circle of radius 2 in the $x z$ -plane, centered at the origin.

Bradley Duda
Bradley Duda
Numerade Educator
View

Problem 24

Find a parameterization for the curve.
The circle of radius 3 parallel to the $x y$ -plane, centered at the point (0,0,2).

Bradley Duda
Bradley Duda
Numerade Educator
View

Problem 25

Find a parameterization for the curve.
The circle of radius 3 in the $y z$ -plane, centered at the point (0,0,2).

Bradley Duda
Bradley Duda
Numerade Educator
View

Problem 26

Find a parameterization for the curve.
The circle of radius 5 parallel to the $y z$ -plane, centered at the point (-1,0,-2).

Bradley Duda
Bradley Duda
Numerade Educator
01:04

Problem 27

Find a parameterization for the curve.
The curve $x=y^{2}$ in the $x y$ -plane.

Khushbu Rani
Khushbu Rani
Numerade Educator
01:09

Problem 28

Find a parameterization for the curve.
The curve $y=x^{3}$ in the $x y$ -plane.

Khushbu Rani
Khushbu Rani
Numerade Educator
01:04

Problem 29

Find a parameterization for the curve.
The curve $x=-3 z^{2}$ in the $x z$ -plane.

Khushbu Rani
Khushbu Rani
Numerade Educator
01:14

Problem 30

Find a parameterization for the curve.
The curve in which the plane $z=2$ cuts the surface $z=\sqrt{x^{2}+y^{2}}$.

WM
William Mead
Numerade Educator
02:05

Problem 31

Find a parameterization for the curve.
The curve $y=4-5 x^{4}$ through the point $(0,4,4),$ parallel to the $x y$ -plane.

William Semus
William Semus
Numerade Educator
01:47

Problem 32

Find a parameterization for the curve.
The ellipse of major diameter 5 parallel to the $y$ -axis and minor diameter 2 parallel to the $z$ -axis, centered at (0,1,-2)

WM
William Mead
Numerade Educator
00:56

Problem 33

Find a parameterization for the curve.
The ellipse of major diameter 6 along the $x$ -axis and minor diameter 4 along the $y$ -axis, centered at the origin.

WM
William Mead
Numerade Educator
01:47

Problem 34

Find a parameterization for the curve.
The ellipse of major diameter 3 parallel to the $x$ -axis and minor diameter 2 parallel to the $z$ -axis, centered at (0,1,-2).

WM
William Mead
Numerade Educator
02:56

Problem 35

Find a parametric equation for the curve segment.
Line from (-1,2,-3) to (2,2,2).

Khushbu Rani
Khushbu Rani
Numerade Educator
02:12

Problem 36

Find a parametric equation for the curve segment.
Line from $P_{0}=(-1,-3)$ to $P_{1}=(5,2)$.

Khushbu Rani
Khushbu Rani
Numerade Educator
01:48

Problem 37

Find a parametric equation for the curve segment.
Line from $P_{0}=(1,-3,2)$ to $P_{1}=(4,1,-3)$.

WM
William Mead
Numerade Educator
02:14

Problem 38

Find a parametric equation for the curve segment.
Semicircle from (0,0,5) to (0,0,-5) in the $y z$ -plane with $y \geq 0$.

WM
William Mead
Numerade Educator
01:15

Problem 39

Find a parametric equation for the curve segment.
Semicircle from (1,0,0) to (-1,0,0) in the $x y$ -plane.

WM
William Mead
Numerade Educator
00:44

Problem 40

Find a parametric equation for the curve segment.
Graph of $y=\sqrt{x}$ from (1,1) to (16,4).

WM
William Mead
Numerade Educator
01:15

Problem 41

Find a parametric equation for the curve segment.
Arc of a circle of radius 5 from $P=(0,0)$ to $Q=$ (10,0).

WM
William Mead
Numerade Educator
02:02

Problem 42

Find a parametric equation for the curve segment.
Quarter-ellipse from (4,0,3) to (0,-3,3) in the plane $z=3$.

Madi Sousa
Madi Sousa
Numerade Educator
01:22

Problem 43

Find parametric equations for a helix satisfying the given conditions.
Centered on the $z$ -axis, with radius 10.

Zack A
Zack A
Numerade Educator
02:41

Problem 44

Find parametric equations for a helix satisfying the given conditions.
Centered on the $x$ -axis, with radius 5.

WM
William Mead
Numerade Educator
01:55

Problem 45

Find parametric equations for a helix satisfying the given conditions.
Centered on the $y$ -axis, with radius 2.

WM
William Mead
Numerade Educator
01:51

Problem 46

Find parametric equations for a helix satisfying the given conditions.
Centered on the vertical line passing through (3,5,0) with radius $1 .$

WM
William Mead
Numerade Educator
01:26

Problem 47

Parameterize the line through $P=(2,5)$ and $Q=(12,9)$ so that the points $P$ and $Q$ correspond to the given parameter values.
$t=0$ and 1

WM
William Mead
Numerade Educator
01:58

Problem 48

Parameterize the line through $P=(2,5)$ and $Q=(12,9)$ so that the points $P$ and $Q$ correspond to the given parameter values.
$t=0$ and 5

WM
William Mead
Numerade Educator
03:11

Problem 49

Parameterize the line through $P=(2,5)$ and $Q=(12,9)$ so that the points $P$ and $Q$ correspond to the given parameter values.
$t=20$ and 30

WM
William Mead
Numerade Educator
03:34

Problem 50

Parameterize the line through $P=(2,5)$ and $Q=(12,9)$ so that the points $P$ and $Q$ correspond to the given parameter values.
$t=10$ and 11

WM
William Mead
Numerade Educator
02:03

Problem 51

Parameterize the line through $P=(2,5)$ and $Q=(12,9)$ so that the points $P$ and $Q$ correspond to the given parameter values.
$t=0$ and -1

WM
William Mead
Numerade Educator
02:32

Problem 52

At the point where $t=-1,$ find an equation for the plane perpendicular to the line
$$x=5-3 t, \quad y=5 t-7, \quad \frac{z}{t}=6$$

WM
William Mead
Numerade Educator
01:44

Problem 53

Determine whether the following line is parallel to the plane $2 x-3 y+5 z=5$:
$$x=5+7 t, \quad y=4+3 t, \quad z=-3-2 t$$

WM
William Mead
Numerade Educator
02:13

Problem 54

Show that the equations $x=3+t, y=2 t, z=1-t$ satisfy the equations $x+y+3 z=6$ and $x-y-z=2$ What does this tell you about the curve parameterized by these equations?

WM
William Mead
Numerade Educator
04:42

Problem 55

(a) Explain why the line of intersection of two planes must be parallel to the cross product of a normal vector to the first plane and a normal vector to the second.
(b) Find a vector parallel to the line of intersection of the two planes $x+2 y-3 z=7$ and $3 x-y+z=0$
(c) Find parametric equations for the line in part (b).

WM
William Mead
Numerade Educator
05:37

Problem 56

Find an equation for the plane containing the point (2,3,4) and the line $x=1+2 t, y=3-t, z=4+t$.

Khushbu Rani
Khushbu Rani
Numerade Educator
04:26

Problem 57

(a) Find an equation for the line perpendicular to the plane $2 x-3 y=z$ and through the point (1,3,7).
(b) Where does the line cut the plane?
(c) What is the distance between the point (1,3,7) and the plane?

WM
William Mead
Numerade Educator
03:33

Problem 58

Consider two points $P_{0}$ and $P_{1}$ in 3 -space.
(a) Show that the line segment from $P_{0}$ to $P_{1}$ can be parameterized by
$$\vec{r}(t)=(1-t) \overrightarrow{O P_{0}}+t \overrightarrow{O P_{1}}, \quad 0 \leq t \leq 1$$
(b) What is represented by the parametric equation
$$\vec{r}(t)=t \overrightarrow{O P_{0}}+(1-t) \overrightarrow{O P_{1}}, \quad 0 \leq t \leq 1 ?$$

WM
William Mead
Numerade Educator
02:06

Problem 59

(a) Find a vector parallel to the line of intersection of the planes $2 x-y-3 z=0$ and $x+y+z=1$.
(b) Show that the point (1,-1,1) lies on both planes.
(c) Find parametric equations for the line of intersection.

WM
William Mead
Numerade Educator
01:58

Problem 60

Find the intersection of the line $x=5+7 t, y=4+3 t$ $z=-3-2 t$ and the plane $2 x-3 y+5 z=-7$.

WM
William Mead
Numerade Educator
02:23

Problem 61

Are the lines $L_{1}$ and $L_{2}$ the same line?
$$\begin{aligned}
&L_{1}: x=5+t, y=3-2 t, z=5 t\\
&L_{2}: x=5+2 t, y=3-4 t, z=10 t
\end{aligned}$$

WM
William Mead
Numerade Educator
01:54

Problem 62

Are the lines $L_{1}$ and $L_{2}$ the same line?
$$\begin{array}{l}
L_{1}: x=2+3 t, y=1+4 t, z=6-t \\
L_{2}: x=2+6 t, y=4+3 t, z=3-2 t
\end{array}$$

WM
William Mead
Numerade Educator
02:26

Problem 63

Are the lines $L_{1}$ and $L_{2}$ the same line?
$$\begin{array}{l}
L_{1}: x=2+3 t, y=1+4 t, z=6-t \\
L_{2}: x=5+6 t, y=5+8 t, z=5-2 t
\end{array}$$

WM
William Mead
Numerade Educator
02:32

Problem 64

Are the lines $L_{1}$ and $L_{2}$ the same line?
$$\begin{array}{l}
L_{1}: x=1+2 t, y=1-3 t, z=1+t \\
L_{2}: x=1-4 t, y=6 t, z=4-2 t
\end{array}$$

WM
William Mead
Numerade Educator
01:58

Problem 65

Two parameterized lines are given. Are they the same line?
$$\begin{aligned}
&\vec{r}_{1}(t)=(5-3 t) \vec{i}+2 t \vec{j}+(7+t) \vec{k}\\
&\vec{r}_{2}(t)=(5-6 t) \vec{i}+4 t \vec{j}+(7+3 t) \vec{k}
\end{aligned}$$

WM
William Mead
Numerade Educator
03:24

Problem 66

Two parameterized lines are given. Are they the same line?
$$\begin{aligned}
&\vec{r}_{1}(t)=(5-3 t) \vec{i}+(1+t) \vec{j}+2 t \vec{k}\\
&\vec{r}_{2}(t)=(2+6 t) \vec{i}+(2-2 t) \vec{j}+(2-4 t) \vec{k}
\end{aligned}$$

WM
William Mead
Numerade Educator
03:27

Problem 67

Two parameterized lines are given. Are they the same line?
$$\begin{array}{l}
\vec{r}_{1}(t)=(5-3 t) \vec{i}+(1+t) \vec{j}+2 t \vec{k} \\
\vec{r}_{2}(t)=(2+6 t) \vec{i}+(2-2 t) \vec{j}+(3-4 t) \vec{k}
\end{array}$$

WM
William Mead
Numerade Educator
02:45

Problem 68

If it exists, find the value of $c$ for which the lines $l(t)=$ $(c+t, 1+t, 5+t)$ and $m(s)=(s, 1-s, 3+s)$ intersect.

WM
William Mead
Numerade Educator
03:15

Problem 69

(a) Where does the line $\vec{r}=2 \vec{i}+5 \vec{j}+t(3 \vec{i}+\vec{j}+2 \vec{k})$ cut the plane $x+y+z=1 ?$
(b) Find a vector perpendicular to the line and lying in the plane.
(c) Find an equation for the line that passes through the point of intersection of the line and plane, is perpendicular to the line, and lies in the plane.

WM
William Mead
Numerade Educator
01:43

Problem 70

Find parametric equations for the line.
The line of intersection of the planes $x-y+z=3$ and $2 x+y-z=5$.

WM
William Mead
Numerade Educator
02:37

Problem 71

Find parametric equations for the line.
The line of intersection of the planes $x+y+z=3$ and $x-y+2 z=2$.

WM
William Mead
Numerade Educator
01:11

Problem 72

Find parametric equations for the line.
The line perpendicular to the surface $z=x^{2}+y^{2}$ at the point (1,2,5).

WM
William Mead
Numerade Educator
01:58

Problem 73

Find parametric equations for the line.
The line through the point (-4,2,3) and parallel to a line in the $y z$ -plane which makes a $45^{\circ}$ angle with the positive $y$ -axis and the positive $z$ -axis.

WM
William Mead
Numerade Educator
05:40

Problem 74

Is the point (-3,-4,2) visible from the point (4,5,0) if there is an opaque ball of radius 1 centered at the origin?

WM
William Mead
Numerade Educator
06:40

Problem 75

Two particles are traveling through space. At time $t$ the first particle is at the point $(-1+t, 4-t,-1+2 t)$ and the second particle is at $(-7+2 t,-6+2 t,-1+t)$
(a) Describe the two paths in words.
(b) Do the two particles collide? If so, when and where?
(c) Do the paths of the two particles cross? If so, where?

WM
William Mead
Numerade Educator
02:52

Problem 76

Match the parameterizations with their graphs in Figure 17.9.
(a) $x=2 \cos 4 \pi t, y=2 \sin 4 \pi t, z=t$
(b) $x=2 \cos 4 \pi t, y=\sin 4 \pi t, z=t$
(c) $x=0.5 t \cos 4 \pi t, y=0.5 t \sin 4 \pi t, z=t$
(d) $x=2 \cos 4 \pi t, y=2 \sin 4 \pi t, z=0.5 t^{3}$
(FIGURE CAN'T COPY)

WM
William Mead
Numerade Educator
01:32

Problem 77

Find $c$ so that one revolution about the $z$ axis of the helix gives an increase of $\Delta z$ in the $z$ -coordinate.
$$x=2 \cos t, y=2 \sin t, z=c t, \Delta z=15$$

WM
William Mead
Numerade Educator
01:22

Problem 78

Find $c$ so that one revolution about the $z$ axis of the helix gives an increase of $\Delta z$ in the $z$ -coordinate.
$$x=2 \cos t, y=2 \sin t, z=c t, \Delta z=50$$

WM
William Mead
Numerade Educator
01:48

Problem 79

Find $c$ so that one revolution about the $z$ axis of the helix gives an increase of $\Delta z$ in the $z$ -coordinate.
$$x=2 \cos 3 t, y=2 \sin 3 t, z=c t, \Delta z=10$$

WM
William Mead
Numerade Educator
01:39

Problem 80

Find $c$ so that one revolution about the $z$ axis of the helix gives an increase of $\Delta z$ in the $z$ -coordinate.
$$x=2 \cos \pi t, y=2 \sin \pi t, z=c t, \Delta z=20$$

WM
William Mead
Numerade Educator
04:28

Problem 81

For $t>0,$ a particle moves along the curve $x=$ $a+b \sin k t, y=a+b \cos k t,$ where $a, b, k$ are positive
constants.
(a) Describe the motion in words.
(b) What is the effect on the curve of the following changes?
(i) Increasing $b$
(ii) Increasing $a$
(iii) Increasing $k$
(iv) Setting $a$ and $b$ equal

WM
William Mead
Numerade Educator
06:03

Problem 82

In the Atlantic Ocean off the coast of Newfoundland, Canada, the temperature and salinity (saltiness) vary throughout the year. Figure 17.10 shows a parametric curve giving the average temperature, $T$ (in $^{\circ} \mathrm{C}$ ) and salinity (in grams of salt per kg of water) for $t$ in months, with $t=1$ corresponding to mid-January $^{1}$.
(a) Why does the parameterized curve form a loop?
(b) When is the water temperature highest?
(c) When is the water saltiest?
(d) Estimate $d T / d t$ at $t=6,$ and give the units. What is the meaning of your answer for seawater?

WM
William Mead
Numerade Educator
03:46

Problem 83

A light shines on the helix of Example 3 on page 886 from far down each axis. Sketch the shadow the helix casts on each of the coordinate planes: $x y, x z,$ and $y z$.

WM
William Mead
Numerade Educator
03:29

Problem 84

The paraboloid $z=x^{2}+y^{2}$ and the plane $z=2 x+4 y+4$ intersect in a curve in 3 -space.
(a) Show that the shadow of the intersection in the $x y-$ plane is a circle and find its center and radius.
(b) Parameterize the circle in the $x y$ -plane.
(c) Parameterize the intersection of the paraboloid and the plane in 3 -space.

WM
William Mead
Numerade Educator
03:56

Problem 85

For a positive constant $a$ and $t \geq 0,$ the parametric equations I-V represent the curves described in (a)-(e). Match each description (a)-(e) with its parametric equations and write an equation involving only $x$ and $y$ for the curve.
(a) Line through the origin.
(b) Line not through the origin.
(c) Hyperbola opening along $x$ -axis.
(d) Circle traversed clockwise.
(e) Circle traversed counterclockwise.
I. $x=a \sin t, y=a \cos t$
II. $x=a \sin t, y=a \sin t$
III. $x=a \cos t, y=a \sin t$
IV. $x=a \cos ^{2} t, y=a \sin ^{2} t$
V. $x=a / \cos t, y=a \tan t$

WM
William Mead
Numerade Educator
01:53

Problem 86

(a) Find a parametric equation for the line through the point (2,1,3) and in the direction of $a \vec{i}+b \vec{j}+c \vec{k}$.
(b) Find conditions on $a, b, c$ so that the line you found in part (a) goes through the origin. Give a reason for your answer.

Nick Johnson
Nick Johnson
Numerade Educator
02:08

Problem 87

Consider the line $x=5-2 t, y=3+7 t, z=4 t$ and the plane $a x+b y+c z=d$. All the following questions have many possible answers. Find values of $a, b, c, d$ such that:
(a) The plane is perpendicular to the line.
(b) The plane is perpendicular to the line and through the point (5,3,0)
(c) The line lies in the plane.

Nick Johnson
Nick Johnson
Numerade Educator
01:37

Problem 88

Explain the significance of the constants $\alpha>0$ and $\beta>0$ in the family of helices given by $\vec{r}=\alpha \cos t \vec{i}+$ $\alpha \sin t \vec{j}+\beta t \vec{k}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:45

Problem 89

Find parametric equations of the line passing through the points (1,2,3),(3,5,7) and calculate the shortest distance from the line to the origin.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:03

Problem 90

Show that for a fixed value of $\theta$, the line parameterized by $x=\cos \theta+t \sin \theta, y=\sin \theta-t \cos \theta$ and $z=t$ lies on the graph of the hyperboloid $x^{2}+y^{2}=z^{2}+1$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:47

Problem 91

A line has equation $\vec{r}=\vec{a}+t \vec{b}$ where $\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}$ and $\vec{a}$ and $\vec{b}$ are constant vectors such that $\vec{a} \neq \overrightarrow{0}, \vec{b} \neq$ $\overrightarrow{0}, \vec{b}$ not parallel or perpendicular to $\vec{a} .$ For each of the planes (a)-(c), pick the equation (i)-(ix) which represents it. Explain your choice.
(a) A plane perpendicular to the line and through the origin.
(b) A plane perpendicular to the line and not through the origin.
(c) A plane containing the line.
(i) $\vec{a} \cdot \vec{r}=|| \vec{b}||$
(ii) $\vec{b} \cdot \vec{r}=|| \vec{a}||$
(iii) $\vec{a} \cdot \vec{r}=\vec{b} \cdot \vec{r}$
(iv) $(\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{a})=0$
(v) $\vec{r}-\vec{a}=\vec{b}$
(vi) $\vec{a} \cdot \vec{r}=0$
(vii) $\vec{b} \cdot \vec{r}=0$
(viii) $\vec{a}+\vec{r}=\vec{b}$
(ix) $(\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{b})=\|\vec{a}\|$

Nick Johnson
Nick Johnson
Numerade Educator
02:16

Problem 92

(a) Find a parametric equation for the line through the point (1,5,2) and in the direction of the vector $2 \vec{i}+3 \vec{j}-\vec{k}$
(b) By minimizing the square of the distance from a point on the line to the origin, find the exact point on the line which is closest to the origin.

Nick Johnson
Nick Johnson
Numerade Educator
03:00

Problem 93

A plane from Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at $650 \mathrm{km} /$ hour at a constant height of 8000 meters above the line joining Denver and Bismark. Bismark is about $850 \mathrm{km}$ in the direction $60^{\circ}$ north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the $x$ -axis points east and the $y$ -axis points north, and that the earth is flat. Measure distances in kilometers and time in hours.

Nick Johnson
Nick Johnson
Numerade Educator
01:53

Problem 94

The vector $\vec{n}$ is perpendicular to the plane $P_{1} .$ The vector $\vec{v}$ is parallel to the line $L$
(a) If $\vec{n} \cdot \vec{v}=0,$ what does this tell you about the directions of $P_{1}$ and $L ?$ (Are they parallel? Perpendicular? Or is it impossible to tell?)
(b) Suppose $\vec{n} \times \vec{v} \neq \overrightarrow{0} .$ The plane $P_{2}$ has normal $\vec{n} \times \vec{v} .$ What can you say about the directions of
(i) $\quad P_{1}$ and $P_{2} ?$
(ii) $\quad L$ and $P_{2} ?$

Nick Johnson
Nick Johnson
Numerade Educator
01:06

Problem 95

Figure 17.11 shows the parametric curve $x=x(t), y=$ $y(t)$ for $a \leq t \leq b$. (FIGURE CAN'T COPY)
(a) Match a graph to each of the parametric curves given, for the same $t$ values, by
(i) $\quad(-x(t),-y(t))$
(ii) $\quad(-x(t), y(t))$
(iii) $\quad(x(t)+1, y(t))$
(iv) $\quad(x(t)+1, y(t)+1)$
(FIGURES CAN'T COPY)
(b) Which of the following could be the formulas for the functions $x(t), y(t) ?$
(i) $x=10 \cos t \quad y=10 \sin t$
(ii) $x=(10+8 t) \cos t \quad y=(10+8 t) \sin t$
(iii) $x=e^{t^{2} / 200} \cos t \quad y=e^{t^{2} / 200} \sin t$
(iv) $x=(10-8 t) \cos t \quad y=(10-8 t) \sin t$
(v) $x=10 \cos \left(t^{2}+t\right) \quad y=10 \sin \left(t^{2}+t\right)$

Carson Merrill
Carson Merrill
Numerade Educator
00:59

Problem 96

Explain what is wrong with the statement.
The curve parameterized by $\vec{r}_{1}(t)=\vec{r}(t-2),$ defined for all $t$, is a shift in the $\vec{i}$ - direction of the curve parameterized by $\vec{r}(t)$.

Nick Johnson
Nick Johnson
Numerade Educator
01:16

Problem 97

Explain what is wrong with the statement.
All points of the curve $\vec{r}(t)=R \cos t \vec{i}+R \sin t \vec{j}+t \vec{k}$ are the same distance, $R,$ from the origin.

Nick Johnson
Nick Johnson
Numerade Educator
01:08

Problem 98

Give an example of:
Parameterizations of two different circles that have the same center and equal radii.

Nick Johnson
Nick Johnson
Numerade Educator
01:30

Problem 99

Give an example of:
Parameterizations of two different lines that intersect at the point (1,2,3).

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:16

Problem 100

Give an example of:
A parameterization of the line $x=t, y=2 t, z=3+4 t$ that is not given by linear functions.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:11

Problem 101

Are the statements true or false? Give reasons for your answer.
The parametric curve $x=3 t+2, y=-2 t$ for $0 \leq t \leq 5$ passes through the origin.

Nick Johnson
Nick Johnson
Numerade Educator
01:04

Problem 102

Are the statements true or false? Give reasons for your answer.
The parametric curve $x=t^{2}, y=t^{4}$ for $0 \leq t \leq 1$ is a parabola.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:09

Problem 103

Are the statements true or false? Give reasons for your answer.
A parametric curve $x=g(t), y=h(t)$ for $a \leq t \leq b$ is always the graph of a function $y=f(x)$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:35

Problem 104

Are the statements true or false? Give reasons for your answer.
The parametric curve $x=(3 t+2)^{2}, y=(3 t+2)^{2}-1$ for $0 \leq t \leq 3$ is a line.

William Semus
William Semus
Numerade Educator
01:01

Problem 105

Are the statements true or false? Give reasons for your answer.
The parametric curve $x=-\sin t, y=-\cos t$ for $0 \leq t \leq 2 \pi$ traces out a unit circle counterclockwise as $t$ increases.

Nick Johnson
Nick Johnson
Numerade Educator
01:00

Problem 106

Are the statements true or false? Give reasons for your answer.
A parameterization of the graph of $y=\ln x$ for $x>0$ is given by $x=e^{t}, y=t$ for $-\infty<t<\infty$.

Nick Johnson
Nick Johnson
Numerade Educator
01:18

Problem 107

Are the statements true or false? Give reasons for your answer.
Both $x=-t+1, y=2 t$ and $x=2 s, y=-4 s+2$ describe the same line.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:16

Problem 108

Are the statements true or false? Give reasons for your answer.
The line of intersection of the two planes $z=x+y$ and $z=1-x-y$ can be parameterized by $x=t, y=$ $\frac{1}{2}-t, z=\frac{1}{2}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:31

Problem 109

Are the statements true or false? Give reasons for your answer.
The two lines given by $x=t, y=2+t, z=3+t$ and $x=2 s, y=1-s, z=s$ do not intersect.

Nick Johnson
Nick Johnson
Numerade Educator
01:01

Problem 110

Are the statements true or false? Give reasons for your answer.
The line parameterized by $x=1, y=2 t, z=3+t$ is parallel to the $x$ -axis.

Nick Johnson
Nick Johnson
Numerade Educator
01:13

Problem 111

Are the statements true or false? Give reasons for your answer.
The equation $\vec{r}(t)=3 t \vec{i}+(6 t+1) \vec{j}$ parameterizes a line.

Nick Johnson
Nick Johnson
Numerade Educator
01:16

Problem 112

Are the statements true or false? Give reasons for your answer.
The lines parameterized by $\vec{r}_{1}(t)=t \vec{i}+(-2 t+1) \vec{j}$ and $\vec{r}_{2}(t)=(2 t+5) \vec{i}+(-t) \vec{j}$ are parallel.

Nick Johnson
Nick Johnson
Numerade Educator