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Calculus

Earl W. Swokowski

Chapter 14

Vectors and Surfaces - all with Video Answers

Educators

Section 1

Vectors in Two Dimensions

03:46

Problem 1

Sketch the position vector of a and find $\|\mathrm{a}\|$.
$$
a=\langle 2,5\rangle
$$

Uma Kumari
Uma Kumari
Numerade Educator
03:50

Problem 2

Sketch the position vector of a and find $\|\mathrm{a}\|$.
$$
a=\langle-4,-7\rangle
$$

Uma Kumari
Uma Kumari
Numerade Educator
03:40

Problem 3

Sketch the position vector of a and find $\|\mathrm{a}\|$.
$$
a=\langle-5,0\rangle
$$

Uma Kumari
Uma Kumari
Numerade Educator
03:32

Problem 4

Sketch the position vector of a and find $\|\mathrm{a}\|$.
$$
\mathbf{a}=-18 \mathbf{j}
$$

Uma Kumari
Uma Kumari
Numerade Educator
03:40

Problem 5

Sketch the position vector of a and find $\|\mathrm{a}\|$.
$$
a=-4 i+5 j
$$

Uma Kumari
Uma Kumari
Numerade Educator
03:38

Problem 6

Sketch the position vector of a and find $\|\mathrm{a}\|$.
$$
a=2 \mathbf{i}-3 \mathbf{j}
$$

Uma Kumari
Uma Kumari
Numerade Educator
03:56

Problem 7

Find $\mathrm{a}+\mathrm{b}, \mathrm{a}-\mathrm{b}, 2 \mathrm{a},-3 \mathrm{~b},$ and $4 \mathrm{a}-5 \mathrm{~b}$.
$\mathbf{a}=\langle 2,-3\rangle, \quad \mathbf{b}=\langle 1,4\rangle$

Ashly Sunny
Ashly Sunny
Numerade Educator
00:55

Problem 8

Find $\mathrm{a}+\mathrm{b}, \mathrm{a}-\mathrm{b}, 2 \mathrm{a},-3 \mathrm{~b},$ and $4 \mathrm{a}-5 \mathrm{~b}$.
$\mathbf{a}=\langle-2,-5\rangle, \quad \mathbf{b}=\langle-3,4\rangle$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:07

Problem 9

Find $\mathrm{a}+\mathrm{b}, \mathrm{a}-\mathrm{b}, 2 \mathrm{a},-3 \mathrm{~b},$ and $4 \mathrm{a}-5 \mathrm{~b}$.
$\mathbf{a}=-\langle 7,-2\rangle, \quad \mathbf{b}=4\langle-2,1\rangle$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:10

Problem 10

Find $\mathrm{a}+\mathrm{b}, \mathrm{a}-\mathrm{b}, 2 \mathrm{a},-3 \mathrm{~b},$ and $4 \mathrm{a}-5 \mathrm{~b}$.
$\mathbf{a}=2\langle 1,5\rangle, \quad \mathbf{b}=-3\langle-1,-4\rangle$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
05:01

Problem 11

Find $\mathrm{a}+\mathrm{b}, \mathrm{a}-\mathrm{b}, 2 \mathrm{a},-3 \mathrm{~b},$ and $4 \mathrm{a}-5 \mathrm{~b}$.
$\mathbf{a}=3 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{b}=-\mathbf{i}+5 \mathbf{j}$

Ashly Sunny
Ashly Sunny
Numerade Educator
05:01

Problem 12

Find $\mathrm{a}+\mathrm{b}, \mathrm{a}-\mathrm{b}, 2 \mathrm{a},-3 \mathrm{~b},$ and $4 \mathrm{a}-5 \mathrm{~b}$.
$\mathbf{a}=-5 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{b}=\mathbf{i}-3 \mathbf{j}$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:00

Problem 13

Use components to express the sum or difference as a scalar multiple of one of the vectors a, $\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e},$ or $\mathrm{f}$ shown in the figure.
$$
a+b
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
00:26

Problem 14

Use components to express the sum or difference as a scalar multiple of one of the vectors a, $\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e},$ or $\mathrm{f}$ shown in the figure.
$$
\mathbf{c}-\mathbf{d}
$$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:00

Problem 15

Use components to express the sum or difference as a scalar multiple of one of the vectors a, $\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e},$ or $\mathrm{f}$ shown in the figure.
$$
\mathbf{b}+\mathbf{e}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
00:26

Problem 16

Use components to express the sum or difference as a scalar multiple of one of the vectors a, $\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e},$ or $\mathrm{f}$ shown in the figure.
$$
f-b
$$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:00

Problem 17

Use components to express the sum or difference as a scalar multiple of one of the vectors a, $\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e},$ or $\mathrm{f}$ shown in the figure.
$$
b+d
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
01:00

Problem 18

Use components to express the sum or difference as a scalar multiple of one of the vectors a, $\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e},$ or $\mathrm{f}$ shown in the figure.
$$
\mathbf{e}+\mathbf{c}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
00:47

Problem 19

Find the vector a in $V,$ that corresponds to $\overrightarrow{P Q}$. Sketch $\overrightarrow{P Q}$ and the position vector for a.
$$
P(1,-4), \quad Q(5,3)
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:13

Problem 20

Find the vector a in $V,$ that corresponds to $\overrightarrow{P Q}$. Sketch $\overrightarrow{P Q}$ and the position vector for a.
$$
P(7,-3), \quad Q(-2,4)
$$

ES
Ellie Sun
Numerade Educator
01:09

Problem 21

Find the vector a in $V,$ that corresponds to $\overrightarrow{P Q}$. Sketch $\overrightarrow{P Q}$ and the position vector for a.
$$
P(2,5), \quad Q(-4,5)
$$

Subhakanta Sahoo
Subhakanta Sahoo
Numerade Educator
01:46

Problem 22

Find the vector a in $V,$ that corresponds to $\overrightarrow{P Q}$. Sketch $\overrightarrow{P Q}$ and the position vector for a.
$$
P(-4,6), \quad Q(-4,-2)
$$

Nidhi Singhi
Nidhi Singhi
Numerade Educator
01:13

Problem 23

Find the vector a in $V,$ that corresponds to $\overrightarrow{P Q}$. Sketch $\overrightarrow{P Q}$ and the position vector for a.
$$
P(-3,-1), Q(6,-4)
$$

ES
Ellie Sun
Numerade Educator
01:27

Problem 24

Find the vector a in $V,$ that corresponds to $\overrightarrow{P Q}$. Sketch $\overrightarrow{P Q}$ and the position vector for a.
$$
P(2,3), \quad Q(-6,0)
$$

Prashansha Kaushik
Prashansha Kaushik
Numerade Educator
03:03

Problem 25

Find a unit vector that has $\mid$ a) the same direction as a and $|b|$ the opposite direction of a.
$$
\mathbf{a}=-8 \mathbf{i}+15 \mathbf{j}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
02:35

Problem 26

Find a unit vector that has $\mid$ a) the same direction as a and $|b|$ the opposite direction of a.
$$
\mathbf{a}=5 \mathbf{i}-3 \mathbf{j}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
02:48

Problem 27

Find a unit vector that has $\mid$ a) the same direction as a and $|b|$ the opposite direction of a.
$$
\mathbf{a}=\langle 2,-5\rangle
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
02:13

Problem 28

Find a unit vector that has $\mid$ a) the same direction as a and $|b|$ the opposite direction of a.
$$
a=\langle 0,6\rangle
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:52

Problem 29

Find a vector that has the same direction as \langle-6,3\rangle and
(a) twice the magnitude
(b) one-half the magnitude

Ashly Sunny
Ashly Sunny
Numerade Educator
00:38

Problem 30

Find a vector that has the opposite direction of $8 \mathbf{i}-5 \mathbf{j}$ and
(a) three times the magnitude
|b) one-third the magnitude

Ashly Sunny
Ashly Sunny
Numerade Educator
03:34

Problem 31

Find a vector of magnitude 6 that has the same direction as $\mathbf{a}=4 \mathbf{i}-7 \mathbf{j}$

Ashly Sunny
Ashly Sunny
Numerade Educator
02:50

Problem 32

Find a vector of magnitude 4 that has the opposite direction of $\mathbf{a}=\langle 2,-5\rangle$.

Ashly Sunny
Ashly Sunny
Numerade Educator
01:12

Problem 33

Find all real numbers $c$ such that |a) $\|c a\|=3,$ (b) $\|c a\|=-3,$ and $|c|\|c a\|=0$.
$$
\mathbf{a}=3 \mathbf{i}-4 \mathbf{j}
$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:12

Problem 34

Find all real numbers $c$ such that |a) $\|c a\|=3,$ (b) $\|c a\|=-3,$ and $|c|\|c a\|=0$.
$$
\mathbf{a}=\langle-5,12\rangle
$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:39

Problem 35

Approximate the horizontal and vertical components of the vector that is described.
A quarterback releases the football with a velocity of $50 \mathrm{ft} / \mathrm{sec}$ at an angle of 35 with the horizontal.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:32

Problem 36

Approximate the horizontal and vertical components of the vector that is described.
A child pulls a sled through the snow by exerting a force of 20 pounds at an angle of 40 with the horizontal.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:30

Problem 37

Approximate the horizontal and vertical components of the vector that is described.
The biceps muscle, in supporting the forearm and a weight held in the hand, exerts a force of 20 pounds. As shown in the figure, the muscle makes an angle of 108 with the forearm.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:31

Problem 38

Approximate the horizontal and vertical components of the vector that is described.
A jet airplane approaches a runway at an angle of 7.5 with the horizontal, traveling at a velocity of $160 \mathrm{mi} / \mathrm{hr}$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:04

Problem 39

An airplane is flying in the direction 150 with an airspeed of $300 \mathrm{mi} / \mathrm{hr}$, and the wind is blowing at $30 \mathrm{mi} / \mathrm{hr}$ in the direction 60 . Approximate the true course and the ground speed of the airplane.

Wendi Zhao
Wendi Zhao
Numerade Educator
02:25

Problem 40

An airplane pilot wishes to maintain a true course in the direction 240 with a ground speed of $400 \mathrm{mi} / \mathrm{hr}$ when the wind is blowing directly north at $50 \mathrm{mi} / \mathrm{hr}$. Find the required airspeed and compass heading.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:01

Problem 41

Two tugboats are towing a large ship into port, as shown in the figure. The larger tug exerts a force of 4000 pounds on its cable, and the smaller tug exerts a force of 3200 pounds on its cable. If the ship is to travel in a straight line from $A$ to $B$, find the angle $\theta$ that the larger tug must make with the line segment $A B$.

Wendi Zhao
Wendi Zhao
Numerade Educator
03:05

Problem 42

Shown in the figure is an apparatus used to simulate gravity conditions on other planets. A rope is attached to an astronaut who maneuvers on an inclined plane that makes an angle of $\theta$ degrees with the horizontal.
|a) If the astronaut weighs 160 pounds, find the $x$ - and $y$ -components of this downward force (see figure for axes).
(b) The $y$ -component in part (a) is the weight of the astronaut relative to the inclined plane. The astronaut would weigh 27 pounds on the moon and 60 pounds on Mars. Approximate each angle 0 (to the nearest $0.01^{\circ}$ ) so that the inclined-plane apparatus will simulate walking on these surfaces.

Wendi Zhao
Wendi Zhao
Numerade Educator
00:54

Problem 43

Find scalars $p$ and $q$ such that
$$
p\langle 3,-1\rangle+q\langle 4,3\rangle=\langle-6,-11\rangle
$$

Shafiq Rehman
Shafiq Rehman
Numerade Educator
00:45

Problem 44

If $\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle$ and $\mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle$ are any nonzero, non-
parallel vectors and if $\mathrm{c}$ is any other vector, prove that there exist scalars $p$ and $q$ such that $\mathrm{c}=p \mathbf{a}+q \mathbf{b}$. Interpret this fact geometrically.

DL
Dr. Michael Lewchuk
Numerade Educator
00:50

Problem 45

Under what conditions is $\|\mathrm{a}+\mathbf{b}\|=\|\mathbf{a}\|+\|\mathbf{b}\|$ ?

Anas Venkitta
Anas Venkitta
Numerade Educator
02:38

Problem 46

(a) Let a $=\left\langle a_{1}, a_{2}\right\rangle$ be any nonzero vector, and let $\overrightarrow{O A}$ be the position vector for a. If $\theta$ is the smallest nonnegative angle from the positive $x$ -axis to $\overrightarrow{O A}$, show that $\mathbf{a}=\|\mathbf{a}\|(\cos \theta \mathbf{i}+\sin \theta \mathbf{j})$.
(b) Show that every unit vector in $V_{2}$ can be expressed in the form $\cos \theta \mathbf{i}+\sin \theta \mathbf{j}$ for some $\theta$

Ankit Singh
Ankit Singh
Numerade Educator
02:54

Problem 47

Let $\mathbf{r}_{0}=\left\langle x_{0}, y_{0}\right\rangle, \mathbf{r}=\langle x, y\rangle,$ and $c>0 .$ Describe the set of all points $P(x, y)$ such that $\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=c$.

Ashly Sunny
Ashly Sunny
Numerade Educator
04:19

Problem 48

Let $\mathbf{r}_{0}=\left\langle x_{0}, y_{0}\right\rangle, \mathbf{r}=\langle x, y\rangle,$ and $\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle \neq \mathbf{0}$. Describe the set of all points $P(x, y)$ such that $\mathbf{r}-\mathbf{r}_{0}$ is a scalar multiple of a.

Ipsita Mandal
Ipsita Mandal
Numerade Educator
03:16

Problem 49

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
$$
\mathbf{a}+(\mathbf{b}+\mathbf{c})=(\mathbf{a}+\mathbf{b})+\mathbf{c}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
02:51

Problem 50

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
$$
a+0=a
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:18

Problem 51

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
$$
0 \mathbf{a}=\mathbf{0}=p \mathbf{0}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:17

Problem 52

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
$$
1 \mathrm{a}=\mathbf{a}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
02:51

Problem 53

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
$$
(p+q) \mathbf{a}=p \mathbf{a}+q \mathbf{a}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:17

Problem 54

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
$$
p(\mathbf{a}-\mathbf{b})=p \mathbf{a}-p \mathbf{b}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:18

Problem 55

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
If $p \mathbf{a}=\mathbf{0}$ and $p \neq 0,$ then $\mathbf{a}=\mathbf{0}$.

Ashly Sunny
Ashly Sunny
Numerade Educator
01:17

Problem 56

Prove the given property if $\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,$ $\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,$ and $p$ and $q$ are real numbers.
If $p \mathbf{a}=0$ and $\mathbf{a} \neq 0,$ then $p=0$.

Ashly Sunny
Ashly Sunny
Numerade Educator