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Single Variable Calculus: Early Transcendentals

James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

Chapter 13

Vector Functions - all with Video Answers

Educators

Section 1

Vector Functions and Space Curves

01:59

Problem 1

Find the domain of the vector function.
$\mathbf{r}(t)=\left\langle\ln (t+1), \frac{t}{\sqrt{9-r^{\prime}}}, 2^{\prime}\right\rangle$

WZ
Wen Zheng
Numerade Educator
01:15

Problem 2

Find the domain of the vector function.
$\mathbf{r}(t)=\cos t \mathbf{i}+\ln t \mathbf{j}+\frac{1}{t-2} \mathbf{k}$

Carson Merrill
Carson Merrill
Numerade Educator
02:37

Problem 3

Find the limit.
$\lim _{t \rightarrow 0}\left(e^{-3 \pi} \mathbf{i}+\frac{t^2}{\sin ^2 t} \mathbf{j}+\cos 2 t \mathbf{k}\right)$

William Semus
William Semus
Numerade Educator
02:22

Problem 4

Find the limit.
$\lim _{t \rightarrow 1}\left(\frac{t^2-t}{t-1} \mathbf{i}+\sqrt{t+8} \mathbf{j}+\frac{\sin \pi t}{\ln t} \mathbf{k}\right)$

WZ
Wen Zheng
Numerade Educator
01:40

Problem 5

Find the limit.
$\lim _{t \rightarrow \infty}\left\langle\frac{1+t^2}{1-t^2}, \tan ^{-1} t, \frac{1-e^{-2 t}}{t}\right\rangle$

WZ
Wen Zheng
Numerade Educator
01:26

Problem 6

Find the limit.
$\lim _{t \rightarrow \infty}\left\langle t e^{-t} \cdot \frac{t^3+t}{2 t^3-1}, t \sin \frac{1}{t}\right\rangle$

James Kiss
James Kiss
Numerade Educator
01:58

Problem 7

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\langle-\cos t, t\rangle$

James Kiss
James Kiss
Numerade Educator
01:08

Problem 8

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\left\langle t^2-1, t\right\rangle$

Carson Merrill
Carson Merrill
Numerade Educator
03:49

Problem 9

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=(3 \sin t, 2 \cos t)$

Melissa Munoz
Melissa Munoz
Numerade Educator
01:40

Problem 10

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=e^{\prime} \mathbf{i}+e^{-1} \mathbf{j}$

James Kiss
James Kiss
Numerade Educator
01:08

Problem 11

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\langle t .2-t .2 t\rangle$

Carson Merrill
Carson Merrill
Numerade Educator
01:27

Problem 12

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\langle\sin \pi t, t, \cos \pi t\rangle$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 13

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\left\langle 3, t, 2-t^2\right\rangle$

Check back soon!

Problem 14

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+\mathbf{k}$

Check back soon!
02:30

Problem 15

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=t^2 \mathbf{i}+t^4 \mathbf{j}+t^6 \mathbf{k}$

James Kiss
James Kiss
Numerade Educator

Problem 16

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $t$ increases.
$\mathbf{r}(t)=\cos t \mathbf{i}-\cos t \mathbf{j}+\sin t \mathbf{k}$

Check back soon!
01:12

Problem 17

Draw the projection of the curve onto the given plane.
$\mathbf{r}(t)=\left\langle t^2, t^3, t^{-3}\right\rangle$, $y z$-plane

James Kiss
James Kiss
Numerade Educator
01:01

Problem 18

Draw the projection of the curve onto the given plane.
$\mathbf{r}(t)=\langle t+1,3 t+1, \cos (t / 2)\rangle$, $x y$-plane

James Kiss
James Kiss
Numerade Educator
01:49

Problem 19

Draw the projections of the curve onto the three coordinate planes. Use these projections to help sketch the curve.
$\mathbf{r}(t)=\langle t, \sin t, 2 \cos t)$

Nick Johnson
Nick Johnson
Numerade Educator
01:20

Problem 20

Draw the projections of the curve onto the three coordinate planes. Use these projections to help sketch the curve.
$\mathbf{r}(t)=\left\langle t, t, t^2\right\rangle$

Nick Johnson
Nick Johnson
Numerade Educator
02:42

Problem 21

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(-2,1,0), \quad Q(5,2,-3)$

Khushbu Rani
Khushbu Rani
Numerade Educator
00:58

Problem 22

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(0,0,0), Q(-7,4,6)$

James Kiss
James Kiss
Numerade Educator
01:24

Problem 23

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(3.5,-1.4,2.1), Q(1.8,0.3,2.1)$

James Kiss
James Kiss
Numerade Educator
07:49

Problem 24

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.
$P(a, b, c), \quad Q(u, v, w)$

Bobby Barnes
Bobby Barnes
University of North Texas
01:16

Problem 25

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.
$x=t \cos t, \quad y=t, \quad z=t \sin t, \quad t \geqslant 0$

James Kiss
James Kiss
Numerade Educator
00:59

Problem 26

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.
$x=\cos t, \quad y=\sin t, \quad z=1 /\left(1+t^2\right)$

James Kiss
James Kiss
Numerade Educator
00:51

Problem 27

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.
$x=t, \quad y=1 /\left(1+t^2\right), \quad z=t^2$

WZ
Wen Zheng
Numerade Educator
00:59

Problem 28

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.
$x=\cos t, \quad y=\sin t, \quad z=\cos 2 t$

James Kiss
James Kiss
Numerade Educator
01:03

Problem 29

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.
$x=\cos 8 t, \quad y=\sin 8 t, \quad z=e^{0.8 t}, \quad t \geq 0$

James Kiss
James Kiss
Numerade Educator
00:59

Problem 30

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.
$x=\cos ^2 t, \quad y=\sin ^2 t, \quad z=t$

James Kiss
James Kiss
Numerade Educator
00:59

Problem 31

Find an equation of the plane that contains the curve with the given vector equation.
$\mathbf{r}(t)=\left\langle t, 4, t^2\right\rangle$

James Kiss
James Kiss
Numerade Educator
01:04

Problem 32

Find an equation of the plane that contains the curve with the given vector equation.
$\mathbf{r}(t)=\left\langle t, t^2, t\right\rangle$

James Kiss
James Kiss
Numerade Educator
00:58

Problem 33

Find an equation of the plane that contains the curve with the given vector equation.
$\mathbf{r}(t)=\langle\sin t, \cos t,-\cos t\rangle$

James Kiss
James Kiss
Numerade Educator
01:11

Problem 34

Find an equation of the plane that contains the curve with the given vector equation.
$\mathbf{r}(t)=\langle 2 t, \sin t, t+1\rangle$

James Kiss
James Kiss
Numerade Educator
02:16

Problem 35

Show that the curve with parametric equations $x=t \cos t$. $y=t \sin t, z=t$ lies on the cone $z^2-x^2+y^2$, and use this fact to help sketch the curve.

Melissa Munoz
Melissa Munoz
Numerade Educator
03:34

Problem 36

Show that the curve with parametric equations $x=\sin t$, $y=\cos t, z=\sin ^2 t$ is the curve of intersection of the surfaces $z=x^2$ and $x^2+y^2=1$. Use this fact to help sketch the curve.

WZ
Wen Zheng
Numerade Educator
01:28

Problem 37

Find three different surfaces that contain the curve

$$
\mathbf{r}(t)=2 t \mathbf{i}+e^t \mathbf{j}+e^{2 t} \mathbf{k}
$$

James Kiss
James Kiss
Numerade Educator
01:10

Problem 38

Find three different surfaces that contain the curve

$$
\mathbf{r}(t)=t^2 \mathbf{i}+\ln t \mathbf{j}+(1 / t) \mathbf{k}
$$

Carson Merrill
Carson Merrill
Numerade Educator
03:08

Problem 39

At what points does the curve $\mathbf{r}(t)=t \mathbf{i}+\left(2 t-t^2\right) \mathbf{k}$ intersect the paraboloid $z=x^2+y^2$ ?

Bobby Barnes
Bobby Barnes
University of North Texas
01:42

Problem 40

At what points does the helix $\mathbf{r}(t)=\langle\sin t, \cos t, t\rangle$ intersect the sphere $x^2+y^2+z^2=5$ ?

WZ
Wen Zheng
Numerade Educator
01:44

Problem 41

Graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\langle\cos t \sin 2 t, \sin t \sin 2 t, \cos 2 t\rangle$

James Kiss
James Kiss
Numerade Educator
01:11

Problem 42

Graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\left\langle t e^t, e^{-t}, t\right)$

James Kiss
James Kiss
Numerade Educator
01:35

Problem 43

Graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\left\langle\sin 3 t \cos t, \frac{1}{4} t, \sin 3 t \sin t\right)$

James Kiss
James Kiss
Numerade Educator
01:28

Problem 44

Graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=(\cos (8 \cos t) \sin t, \sin (8 \cos t) \sin t, \cos t)$

James Kiss
James Kiss
Numerade Educator
00:59

Problem 45

Graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
$\mathbf{r}(t)=\langle\cos 2 t, \cos 3 t, \cos 4 t\rangle$

James Kiss
James Kiss
Numerade Educator
01:30

Problem 46

Graph the curve with parametric equations

$$
x=\sin t \quad y=\sin 2 t \quad z=\cos 4 t
$$

Explain its shape by graphing its projections onto the three coordinate planes.

WZ
Wen Zheng
Numerade Educator
01:13

Problem 47

Graph the curve with parametric equations

$$
\begin{aligned}
& x=(1+\cos 16 t) \cos t \\
& y=(1+\cos 16 t) \sin t \\
& z=1+\cos 16 t
\end{aligned}
$$

Explain the appearance of the graph by showing that it lies on a conc.

Carson Merrill
Carson Merrill
Numerade Educator
01:13

Problem 48

Graph the curve with parametric equations

$$
\begin{aligned}
& x=\sqrt{1-0.25 \cos ^2 10 t} \cos t \\
& y=\sqrt{1-0.25 \cos ^2 10 t} \sin t \\
& z=0.5 \cos 10 t
\end{aligned}
$$

Explain the appearance of the graph by showing that it lies on a sphere.

Carson Merrill
Carson Merrill
Numerade Educator
01:42

Problem 49

Show that the curve with parametric equations $x=t^2$, $y=1-3 t, z=1+t^3$ passes through the points $(1,4,0)$ and $(9,-8,28)$ but not through the point $(4,7,-6)$.

Aman Gupta
Aman Gupta
Numerade Educator
01:06

Problem 50

Find a vector function that represents the curve of intersection of the two surfaces.
The cylinder $x^2+y^2=4$ and the surface $z=x y$

James Kiss
James Kiss
Numerade Educator
01:34

Problem 51

Find a vector function that represents the curve of intersection of the two surfaces.
The cone $z=\sqrt{x^2+y^2}$ and the plane $z=1+y$

Carson Merrill
Carson Merrill
Numerade Educator
02:41

Problem 52

Find a vector function that represents the curve of intersection of the two surfaces.
The paraboloid $z=4 x^2+y^2$ and the parabolic cylinder $y=x^2$

William Semus
William Semus
Numerade Educator
01:03

Problem 53

Find a vector function that represents the curve of intersection of the two surfaces.
The hyperbolic paraboloid $z=x^2-y^2$ and the cylinder $x^2+y^2=1$

James Kiss
James Kiss
Numerade Educator
01:05

Problem 54

Find a vector function that represents the curve of intersection of the two surfaces.
The semiellipsoid $x^2+y^2+4 z^2=4, y \geqslant 0$, and the cylinder $x^2+z^2=1$

Carson Merrill
Carson Merrill
Numerade Educator
03:12

Problem 55

Try to sketch by hand the curve of intersection of the circular cylinder $x^2+y^2=4$ and the parabolic cylinder $z=x^2$. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

WZ
Wen Zheng
Numerade Educator
03:27

Problem 56

Try to sketch by hand the curve of intersection of the parabolic cylinder $y=x^2$ and the top half of the ellipsoid $x^2+4 y^2+4 z^2=16$. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

WZ
Wen Zheng
Numerade Educator
02:07

Problem 57

Intersection and Collision If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) Their paths might intersect, but we need to know whether the objects are in the same position at the same time. (See Exercises 10.1.55-57.)
The trajectories of two particles are given by the vector functions

$$
\mathbf{r}_1(t)=\left\langle t^2, 7 t-12, t^2\right\rangle \quad \mathbf{r}_2(t)=\left\langle 4 t-3, t^2, 5 t-6\right\rangle
$$

for $t \geqslant 0$. Do the particles collide?

James Kiss
James Kiss
Numerade Educator
00:58

Problem 58

Intersection and Collision If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) Their paths might intersect, but we need to know whether the objects are in the same position at the same time. (See Exercises 10.1.55-57.)
Two particles travel along the space curves

$$
\mathbf{r}_1(t)=\left\langle t, t^2, t^3\right\rangle \quad \mathbf{r}_2(t)=\langle 1+2 t, 1+6 t, 1+14 t\rangle
$$

Do the particles collide? Do their paths intersect?

James Kiss
James Kiss
Numerade Educator
01:41

Problem 59

(a) Graph the curve with parametric equations

$$
\begin{aligned}
& x=\frac{27}{26} \sin 8 t-\frac{8}{15} \sin 18 t \\
& y=-\frac{27}{26} \cos 8 t+\frac{2}{39} \cos 18 t \\
& z=\frac{144}{65} \sin 5 t
\end{aligned}
$$

(b) Show that the curve lies on the hyperboloid of one sheet $144 x^2+144 y^2-25 z^2=100$.

Carson Merrill
Carson Merrill
Numerade Educator
01:11

Problem 60

Trefoil Knot The view of the trefoil knot shown in Figure 9 is accurate, but it doesn't reveal the whole story. Use the parametric equations

$$
\begin{aligned}
& x=(2+\cos 1.5 t) \cos t \\
& y=(2+\cos 1.5 t) \sin t \\
& z=\sin 1.5 t
\end{aligned}
$$

to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the $x y$-plane has polar coordinates $r=2+\cos 1.5 t$ and $\theta=t$, so $r$ varies between 1 and 3. Then show that z has maximum and minimum values when the projection is halfway between $r=1$ and $r=3$.

When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then plot the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)

Carson Merrill
Carson Merrill
Numerade Educator
01:11

Problem 61

Properties of Limits Suppose $\mathbf{u}$ and $\mathbf{v}$ are vector functions that possess limits as $t \rightarrow a$ and let $c$ be a constant. Prove the following properties of limits.
(a) $\lim _{t \rightarrow a}[\mathbf{u}(t)+\mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t)+\lim _{t \rightarrow a} \mathbf{v}(t)$
(b) $\lim _{t \rightarrow a} c \mathbf{u}(t)=c \lim _{t \rightarrow a} \mathbf{u}(t)$
(c) $\lim _{t \rightarrow a}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \cdot \lim _{t \rightarrow a} \mathbf{v}(t)$
(d) $\lim _{t \rightarrow a}[\mathbf{u}(t) \times \mathbf{v}(t)]-\lim _{t \rightarrow a} \mathbf{u}(t) \times \lim _{t \rightarrow a} \mathbf{v}(t)$

Carson Merrill
Carson Merrill
Numerade Educator
01:48

Problem 62

Show that $\lim _{1, \ldots} \mathbf{r}(t)=\mathbf{b}$ if and only if for every $\varepsilon>0$ there is a number $\delta>0$ such that if $0<|t-a|<\delta$ then $|\mathbf{r}(t)-\mathbf{b}|<\varepsilon$

Carson Merrill
Carson Merrill
Numerade Educator