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Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

Chapter 9

Vectors in Two and Three Dimensions - all with Video Answers

Educators

Section 1

Vectors in Two Dimensions

04:53

Problem 1

(a) A vector in the plane is a line segment with an assigned direction. In Figure I below, the vector u has initial point ______ and terminal point ______. Sketch the vectors $2 \mathbf{u}$ and $\mathbf{u}+\mathbf{v}$
(b) A vector in a coordinate plane is expressed by using components. In Figure II below, the vector u has initial point (____, ____ ) and terminal point (____, ____ ) In component form we write $\mathbf{u}=$ (____, ____ ), and $\mathbf{v}=$ (____, ____ ) Then $2 \mathbf{u}=$ (____, ____ ) and $\mathbf{u}+\mathbf{v}=$ (____, ____ )

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:59

Problem 2

(a) The length of a vector $\mathbf{w}=\left\langle a_{1}, a_{2}\right\rangle$ is $|\mathbf{w}|=$ ______ so the length of the vector $\mathbf{u}$ in Figure II is $|\mathbf{u}|=$ _______.
(b) If we know the length $|\mathbf{w}|$ and direction $\theta$ of a vector $\mathbf{w}$, then we can express the vector in component form as $\mathbf{w}=$ (______, _______).

Griffin Goodwin
Griffin Goodwin
Numerade Educator
00:57

Problem 3

Sketching Vectors Sketch the vector indicated. (The vectors $\mathbf{u}$ and $\mathbf{v}$ are shown in the figure.)
$$
2 \mathbf{u}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:12

Problem 4

Sketching Vectors Sketch the vector indicated. (The vectors $\mathbf{u}$ and $\mathbf{v}$ are shown in the figure.)
$$
-\mathbf{v}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:03

Problem 5

Sketching Vectors Sketch the vector indicated. (The vectors $\mathbf{u}$ and $\mathbf{v}$ are shown in the figure.)
$$
\mathbf{u}+\mathbf{v}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:18

Problem 6

Sketching Vectors Sketch the vector indicated. (The vectors $\mathbf{u}$ and $\mathbf{v}$ are shown in the figure.)
$$
\mathbf{u}-\mathbf{v}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:23

Problem 7

Sketching Vectors Sketch the vector indicated. (The vectors $\mathbf{u}$ and $\mathbf{v}$ are shown in the figure.)
$$
\mathbf{v}-2 \mathbf{u}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:11

Problem 8

Sketching Vectors Sketch the vector indicated. (The vectors $\mathbf{u}$ and $\mathbf{v}$ are shown in the figure.)
$$
2 \mathbf{u}+\mathbf{v}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:06

Problem 9

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:07

Problem 10

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:19

Problem 11

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:08

Problem 12

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:03

Problem 13

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.
$$
P(3,2), \quad Q(8,9)
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:15

Problem 14

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.
$$
P(1,1), \quad Q(9,9)
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
00:58

Problem 15

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.
$$
P(5,3), \quad Q(1,0)
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:09

Problem 16

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.
$$
P(-1,3), \quad Q(-6,-1)
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:18

Problem 17

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.
$$
P(-1,-1), \quad Q(-1,1)
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:13

Problem 18

Component Form of Vectors Express the vector with initial point $P$ and terminal point $Q$ in component form.
$$
P(-8,-6), \quad Q(-1,-1)
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:46

Problem 19

Sketching Vectors Sketch the given vector with initial point $(4,3),$ and find the terminal point.
$$
\mathbf{u}=\langle 2,4\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:39

Problem 20

Sketching Vectors Sketch the given vector with initial point $(4,3),$ and find the terminal point.
$$
\mathbf{u}=\langle- 1,2\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:35

Problem 21

Sketching Vectors Sketch the given vector with initial point $(4,3),$ and find the terminal point.
$$
\mathbf{u}=\langle 4,-3\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:25

Problem 22

Sketching Vectors Sketch the given vector with initial point $(4,3),$ and find the terminal point.
$$
\mathbf{u}=\langle- 8,-1\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:33

Problem 23

Sketching Vectors Sketch representations of the given vector with initial points at $(0,0),(2,3),$ and $(-3,5) .$
$$
\mathbf{u}=\langle 3,5\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:35

Problem 24

Sketching Vectors Sketch representations of the given vector with initial points at $(0,0),(2,3),$ and $(-3,5) .$
$$
\mathbf{u}=\langle 4,-6\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:26

Problem 25

Sketching Vectors Sketch representations of the given vector with initial points at $(0,0),(2,3),$ and $(-3,5) .$
$$
\mathbf{u}=\langle- 7,2\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:31

Problem 26

Sketching Vectors Sketch representations of the given vector with initial points at $(0,0),(2,3),$ and $(-3,5) .$
$$
\mathbf{u}=\langle 0,-9\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
00:41

Problem 27

Writing vectors in Terms of i and j Write the given vector in terms of i and $\mathbf{j}$.
$$
\mathbf{u}=\langle 1,4\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
00:46

Problem 28

Writing vectors in Terms of i and j Write the given vector in terms of i and $\mathbf{j}$.
$$
\mathbf{u}=\langle- 2,10\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
00:43

Problem 29

Writing vectors in Terms of i and j Write the given vector in terms of i and $\mathbf{j}$.
$$
\mathbf{u}=\langle 3,0\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
00:50

Problem 30

Writing vectors in Terms of i and j Write the given vector in terms of i and $\mathbf{j}$.
$$
\mathbf{u}=\langle 0,-5\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:57

Problem 31

Operations with Vectors Find $2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},$ and $3 \mathbf{u}-4 \mathbf{v}$ for the given vectors $\mathbf{u}$ and $\mathbf{v} .$
$$
\mathbf{u}=\langle 2,7\rangle, \quad \mathbf{v}=\langle 3,1\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:44

Problem 32

Operations with Vectors Find $2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},$ and $3 \mathbf{u}-4 \mathbf{v}$ for the given vectors $\mathbf{u}$ and $\mathbf{v} .$
$$
\mathbf{u}=\langle- 2,5\rangle, \quad \mathbf{v}=\langle 2,-8\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:47

Problem 33

Operations with Vectors Find $2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},$ and $3 \mathbf{u}-4 \mathbf{v}$ for the given vectors $\mathbf{u}$ and $\mathbf{v} .$
$$
\mathbf{u}=\langle 0,-1\rangle, \quad \mathbf{v}=\langle- 2,0\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:39

Problem 34

Operations with Vectors Find $2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},$ and $3 \mathbf{u}-4 \mathbf{v}$ for the given vectors $\mathbf{u}$ and $\mathbf{v} .$
$$
\mathbf{u}=\mathbf{i}, \quad \mathbf{v}=-2 \mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:40

Problem 35

Operations with Vectors Find $2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},$ and $3 \mathbf{u}-4 \mathbf{v}$ for the given vectors $\mathbf{u}$ and $\mathbf{v} .$
$$
\mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:44

Problem 36

Operations with Vectors Find $2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},$ and $3 \mathbf{u}-4 \mathbf{v}$ for the given vectors $\mathbf{u}$ and $\mathbf{v} .$
$$
\mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
05:34

Problem 37

Magnitude of Vectors Find $|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,$
$|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,$ and $|\mathbf{u}|-|\mathbf{v}|$
$$
\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
05:34

Problem 38

Magnitude of Vectors Find $|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,$
$|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,$ and $|\mathbf{u}|-|\mathbf{v}|$
$$
\mathbf{u}=-2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=\mathbf{i}-2 \mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
04:22

Problem 39

Magnitude of Vectors Find $|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,$
$|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,$ and $|\mathbf{u}|-|\mathbf{v}|$
$$
\mathbf{u}=\langle 10,-1\rangle, \quad \mathbf{v}=\langle- 2,-2\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
04:51

Problem 40

Magnitude of Vectors Find $|\mathbf{u}|,|\mathbf{v}|,|2 \mathbf{u}|,\left|\frac{1}{2} \mathbf{v}\right|,$
$|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|,$ and $|\mathbf{u}|-|\mathbf{v}|$
$$
\mathbf{u}=\langle- 6,6\rangle, \quad \mathbf{v}=\langle- 2,-1\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:55

Problem 41

Components of a vector Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and $\mathbf{j}$.
$$
|\mathbf{v}|=40, \quad \theta=30^{\circ}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:30

Problem 42

Components of a vector Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and $\mathbf{j}$.
$$
|\mathbf{v}|=50, \quad \theta=120^{\circ}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:42

Problem 43

Components of a vector Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and $\mathbf{j}$.
$$
|\mathbf{v}|=1, \quad \theta=225^{\circ}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:05

Problem 44

Components of a vector Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and $\mathbf{j}$.
$$
|\mathbf{v}|=800, \quad \theta=125^{\circ}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:27

Problem 45

Components of a vector Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and $\mathbf{j}$.
$$
|\mathbf{v}|=4, \quad \theta=10^{\circ}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:46

Problem 46

Components of a vector Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and $\mathbf{j}$.
$$
|\mathbf{v}|=\sqrt{3}, \quad \theta=300^{\circ}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:34

Problem 47

Magnitude and Direction of a Vector Find the magnitude and direction (in degrees) of the vector.
$$
\mathbf{v}=\langle 3,4\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:25

Problem 48

Magnitude and Direction of a Vector Find the magnitude and direction (in degrees) of the vector.
$$
\mathbf{v}=\left\langle-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:40

Problem 49

Magnitude and Direction of a Vector Find the magnitude and direction (in degrees) of the vector.
$$
\mathbf{v}=\langle- 12,5\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:07

Problem 50

Magnitude and Direction of a Vector Find the magnitude and direction (in degrees) of the vector.
$$
\mathbf{v}=\langle 40,9\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:13

Problem 51

Magnitude and Direction of a Vector Find the magnitude and direction (in degrees) of the vector.
$$
\mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:52

Problem 52

Magnitude and Direction of a Vector Find the magnitude and direction (in degrees) of the vector.
$$
\mathbf{v}=\mathbf{i}+\mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:17

Problem 53

Components of a Force A man pushes a lawn mower with a force of 30 lb exerted at an angle of $30^{\circ}$ to the ground. Find the horizontal and vertical components of the force.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
View

Problem 54

Components of a Velocity A jet is flying in a direction $\mathrm{N} 20^{\circ} \mathrm{E}$ with a speed of $500 \mathrm{mi} / \mathrm{h}$. Find the north and east components of the velocity.

Danielle Fairburn
Danielle Fairburn
Numerade Educator
02:21

Problem 55

Velocity A river flows due south at $3 \mathrm{mi} / \mathrm{h}$. A swimmer attempting to cross the river heads due east swimming at $2 \mathrm{mi} / \mathrm{h}$ relative to the water. Find the true velocity of the swimmer as a vector.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:11

Problem 56

Velocity Suppose that in Exercise 55 the current is flowing at $1.2 \mathrm{mi} / \mathrm{h}$ due south. In what direction should the swimmer head in order to arrive at a landing point due east of his starting point?

Griffin Goodwin
Griffin Goodwin
Numerade Educator
04:28

Problem 57

Velocity The speed of an airplane is $300 \mathrm{mi} / \mathrm{h}$ relative to the air. The wind is blowing due north with a speed of $30 \mathrm{mi} / \mathrm{h}$. In what direction should the airplane head in order to arrive at a point due west of its location?

Griffin Goodwin
Griffin Goodwin
Numerade Educator
04:36

Problem 58

Velocity A migrating salmon heads in the direction N $45^{\circ} \mathrm{E}$, swimming at $5 \mathrm{mi} / \mathrm{h}$ relative to the water. The prevailing ocean currents flow due east at $3 \mathrm{mi} / \mathrm{h}$. Find the true velocity of the fish as a vector.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
05:47

Problem 59

True Velocity of a Jet A pilot heads his jet due east. The jet has a speed of $425 \mathrm{mi} / \mathrm{h}$ relative to the air. The wind is blowing due north with a speed of $40 \mathrm{mi} / \mathrm{h}$.
(a) Express the velocity of the wind as a vector in component form.
(b) Express the velocity of the jet relative to the air as a vector in component form.
(c) Find the true velocity of the jet as a vector.
(d) Find the true speed and direction of the jet.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
09:07

Problem 60

True Velocity of a Jet A jet is flying through a wind that is blowing with a speed of $55 \mathrm{mi} / \mathrm{h}$ in the direction $\mathrm{N} 30^{\circ} \mathrm{E}$ (see the figure). The jet has a speed of $765 \mathrm{mil}$ h relative to the air, and the pilot heads the jet in the direction $\mathrm{N} 45^{\circ} \mathrm{E}$.
(a) Express the velocity of the wind as a vector in component form.
(b) Express the velocity of the jet relative to the air as a vector in component form.
(c) Find the true velocity of the jet as a vector.
(d) Find the true speed and direction of the jet.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
07:07

Problem 61

True Velocity of a Jet Find the true speed and direction of the jet in Exercise 60 if the pilot heads the plane in the direction $\mathrm{N} 30^{\circ} \mathrm{W}$.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
04:33

Problem 62

True Velocity of a Jet In what direction should the pilot in Exercise 60 head the plane for the true course to be due north?

Griffin Goodwin
Griffin Goodwin
Numerade Educator
06:45

Problem 63

Velocity of a Boat A straight river flows east at a speed of $10 \mathrm{mi} / \mathrm{h} .$ A boater starts at the south shore of the river and heads in a direction $60^{\circ}$ from the shore (see the figure). The motorboat has a speed of $20 \mathrm{mi} / \mathrm{h}$ relative to the water.
(a) Express the velocity of the river as a vector in component form.
(b) Express the velocity of the motorboat relative to the water as a vector in component form.
(c) Find the true velocity of the motorboat.
(d) Find the true speed and direction of the motorboat.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:21

Problem 64

Velocity of a Boat The boater in Exercise 63 wants to arrive at a point on the north shore of the river directly opposite the starting point. In what direction should the boat be headed?

Griffin Goodwin
Griffin Goodwin
Numerade Educator
04:45

Problem 65

Velocity of a Boat $A$ boat heads in the direction $\mathrm{N} 72^{\circ} \mathrm{E}$. The speed of the boat relative to the water is $24 \mathrm{mi} / \mathrm{h}$. The water is flowing directly south. It is observed that the true direction of the boat is directly east.
(a) Express the velocity of the boat relative to the water as a vector in component form.
(b) Find the speed of the water and the true speed of the boat.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:47

Problem 66

Velocity A woman walks due west on the deck of an ocean liner at $2 \mathrm{mi} / \mathrm{h}$. The ocean liner is moving due north at a speed of $25 \mathrm{mi} / \mathrm{h}$. Find the speed and direction of the woman relative to the surface of the water.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:21

Problem 67

Equilibrium of Forces The forces $\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}$ acting at the same point $P$ are said to be in equilibrium if the resultant force is zero, that is $\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=\mathbf{0} .$ Find (a) the resultant forces acting at $P,$ and (b) the additional force required (if any) for the forces to be in equilibrium.
$$
\mathbf{F}_{1}=\langle 2,5\rangle, \quad \mathbf{F}_{2}=\langle 3,-8\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:03

Problem 68

Equilibrium of Forces The forces $\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}$ acting at the same point $P$ are said to be in equilibrium if the resultant force is zero, that is $\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=\mathbf{0} .$ Find (a) the resultant forces acting at $P,$ and (b) the additional force required (if any) for the forces to be in equilibrium.
$$
\mathbf{F}_{1}=\langle 3,-7\rangle, \quad \mathbf{F}_{2}=\langle 4,-2\rangle, \quad \mathbf{F}_{3}=\langle- 7,9\rangle
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:27

Problem 69

Equilibrium of Forces The forces $\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}$ acting at the same point $P$ are said to be in equilibrium if the resultant force is zero, that is $\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=\mathbf{0} .$ Find (a) the resultant forces acting at $P,$ and (b) the additional force required (if any) for the forces to be in equilibrium.
$$
\begin{array}{l}{\mathbf{F}_{1}=4 \mathbf{i}-\mathbf{j}, \quad \mathbf{F}_{2}=3 \mathbf{i}-7 \mathbf{j}, \quad \mathbf{F}_{3}=-8 \mathbf{i}+3 \mathbf{j}} \\ {\mathbf{F}_{4}=\mathbf{i}+\mathbf{j}}\end{array}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
01:00

Problem 70

Equilibrium of Forces The forces $\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}$ acting at the same point $P$ are said to be in equilibrium if the resultant force is zero, that is $\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=\mathbf{0} .$ Find (a) the resultant forces acting at $P,$ and (b) the additional force required (if any) for the forces to be in equilibrium.
$$
\mathbf{F}_{1}=\mathbf{i}-\mathbf{j}, \quad \mathbf{F}_{2}=\mathbf{i}+\mathbf{j}, \quad \mathbf{F}_{3}=-2 \mathbf{i}+\mathbf{j}
$$

Griffin Goodwin
Griffin Goodwin
Numerade Educator
04:51

Problem 71

Equilibrium of Forces The forces $\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}$ acting at the same point $P$ are said to be in equilibrium if the resultant force is zero, that is $\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=\mathbf{0} .$ Find (a) the resultant forces acting at $P,$ and (b) the additional force required (if any) for the forces to be in equilibrium.
(graph can't copy)

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:58

Problem 72

Equilibrium of Forces The forces $\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}$ acting at the same point $P$ are said to be in equilibrium if the resultant force is zero, that is $\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=\mathbf{0} .$ Find (a) the resultant forces acting at $P,$ and (b) the additional force required (if any) for the forces to be in equilibrium.
(graph can't copy)

Griffin Goodwin
Griffin Goodwin
Numerade Educator
06:22

Problem 73

Equilibrium of Tensions $A 100-$ lb weight hangs from a string as shown in the figure. Find the tensions $\mathbf{T}_{1}$ and $\mathbf{T}_{2}$ in the string.
(figure can't copy)

Griffin Goodwin
Griffin Goodwin
Numerade Educator
06:19

Problem 74

Equilibrium of Tensions The cranes in the figure are lifting an object that weighs $18,278$ lb. Find the tensions $\mathbf{T}_{1}$ and $\mathbf{T}_{2} .$
(figure can't copy)

Griffin Goodwin
Griffin Goodwin
Numerade Educator
02:07

Problem 75

Discuss: Vectors That Form a Polygon Suppose that $n$ vectors can be placed head to tail in the plane so that they form a polygon. (The figure shows the case of a hexagon.) Explain why the sum of these vectors is 0 .
(figure can't copy)

Griffin Goodwin
Griffin Goodwin
Numerade Educator