Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 3

Vectors - all with Video Answers

Educators

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Chapter Questions

Problem 1

What are (a) the $x$ component and (b) the $y$ component of a vector $\vec{a}$ in the $x y$ plane if its direction is $250^{\circ} \mathrm{y}$ counterclockwise from the positive direction of the $x$ axis and its magnitude is $7.3 \mathrm{~m}$ ?

Anand Jangid

Anand Jangid

Numerade Educator

Problem 2

A displacement vector $\vec{r}$ in the $x y$ plane is $15 \mathrm{~m}$ long and directed at angle $\theta=30^{\circ}$ in Fig. 3 -26. Determine (a) the $x$ component and (b) the $y$ component of the vector.

Derek Walkama

Numerade Educator

Problem 3

The $x$ component of vector $\vec{A}$ is $-25.0 \mathrm{~m}$ and the $y$ component is $+40.0 \mathrm{~m}$. (a) What is the magnitude of $\vec{A} ?$ (b) What is the angle between the direction of $\vec{A}$ and the positive direction of $x ?$

Molika So

University of North Florida

Problem 4

Express the following angles in radians: (a) $20.0^{\circ}$, (b) $50.0^{\circ}$,
(c) $100^{\circ}$. Convert the following angles to degrees:
(d) $0.330$ rad,
(e) $2.10 \mathrm{rad},(\mathrm{f}) 7.70 \mathrm{rad}$.

Derek Walkama

Numerade Educator

Problem 5

A ship sets out to sail to a point $120 \mathrm{~km}$ due north. An unexpected storm blows the ship to a point $100 \mathrm{~km}$ due east of its starting point. (a) How far and (b) in what direction must it now sail to reach its original destination?

Molika So

University of North Florida

Problem 6

In Fig. $3-27$, a heavy piece of machinery is raised by sliding it a distance $d=12.5 \mathrm{~m}$ along a plank oriented at angle $\theta=20.0^{\circ}$ to the horizontal. How far is it moved
(a) vertically and
(b) horizontally?

Derek Walkama

Numerade Educator

Problem 7

Consider two displacements, one of magnitude $3 \mathrm{~m}$ and another of magnitude $4 \mathrm{~m}$. Show how the displacement vectors may be combined to get a resultant displacement of magnitude (a) $7 \mathrm{~m}$, (b) $1 \mathrm{~m}$, and (c) $5 \mathrm{~m}$.

Zachary Warner

Numerade Educator

Problem 8

A person walks in the following pattern: $3.1 \mathrm{~km}$ north, then $2.4 \mathrm{~km}$ west, and finally $5.2 \mathrm{~km}$ south. (a) Sketch the vector diagram that represents this motion. (b) How far and (c) in what direction would a bird fly in a straight line from the same starting point to the -8 A person walks in the following pattern: $3.1 \mathrm{~km}$ north, then $2.4 \mathrm{~km}$ west, and finally $5.2 \mathrm{~km}$ south. (a) Sketch the vector diagram that represents this motion. (b) How far and (c) in what direction would a bird fly in a straight line from the same starting point to the same final point?

Keshav Singh

Numerade Educator

Problem 9

Two vectors are given by
and
$$\begin{aligned}&\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}-(3.0 \mathrm{~m}) \hat{\mathrm{j}}+(1.0 \mathrm{~m}) \hat{\mathrm{k}} \\
&\vec{b}=(-1.0 \mathrm{~m}) \hat{\mathrm{i}}+(1.0 \mathrm{~m}) \hat{\mathrm{j}}+(4.0 \mathrm{~m}) \hat{\mathrm{k}}\end{aligned}$$
In unit-vector notation, find (a) $\vec{a}+\vec{b}$, (b) $\vec{a}-\vec{b}$, and $(\mathrm{c})$ a third vector $\vec{c}$ such that $\vec{a}-\vec{b}+\vec{c}=0$

Zachary Warner

Numerade Educator

Problem 10

Find the (a) $x$, (b) $y$, and (c) $z$ components of the sum $\vec{r}$ of the displacements $\vec{c}$ and $\vec{d}$ whose components in meters are $c_{x}=7.4, c_{y}=-3.8, c_{z}=-6.1 ; d_{x}=4.4, d_{y}=-2.0, d_{z}=3.3$
$-11$ ssM (a) In unit-vector notation, what is the sum $\vec{a}+\vec{b}$ if $\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}+(3.0 \mathrm{~m}) \hat{\mathrm{j}}$ and $\vec{b}=(-13.0 \mathrm{~m}) \hat{\mathrm{i}}+(7.0 \mathrm{~m}) \hat{\mathrm{j}} ?$ What
are the (b) magnitude and (c) direction of $\vec{a}+\vec{b}$ ?

Keshav Singh

Numerade Educator

Problem 11

(a) In unit-vector notation, what is the sum $\vec{a}+\vec{b}$ if $\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}+(3.0 \mathrm{~m}) \hat{\mathrm{j}}$ and $\vec{b}=(-13.0 \mathrm{~m}) \hat{\mathrm{i}}+(7.0 \mathrm{~m}) \hat{\mathrm{j}} ? \quad$ What
are the (b) magnitude and (c) direction of $\vec{a}+\vec{b}$ ?

Molika So

University of North Florida

Problem 12

A car is driven east for a distance of $50 \mathrm{~km}$, then north for 30 $\mathrm{km}$, and then in a direction $30^{\circ}$ east of north for $25 \mathrm{~km}$. Sketch the vector diagram and determine (a) the magnitude and (b) the angle of the car's total displacement from its starting point.

Derek Walkama

Numerade Educator

Problem 13

A person desires to reach a point that is $3.40 \mathrm{~km}$ from her present location and in a direction that is $35.0^{\circ}$ north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?

Zachary Warner

Numerade Educator

Problem 14

You are to make four straight-line moves over a flat desert floor, starting at the origin of an $x y$ coordinate system and ending at the $x y$ coordinates $(-140 \mathrm{~m}, 30 \mathrm{~m})$. The $x$ component and $y$ component of your moves are the following, respectively, in meters: $(20$ and 60$)$, then $\left(b_{x}\right.$ and $\left.-70\right)$, then $\left(-20\right.$ and $\left.c_{y}\right)$, then $(-60$ and $-70$ ). What are (a) component $b_{x}$ and (b) component $c_{y}$ ? What are (c) the magnitude and (d) the angle (relative to the positive direction of the $x$ axis) of the overall displacement?

Derek Walkama

Numerade Educator

Problem 15

The two vectors $\vec{a}$ and $\vec{b}$ in Fig. $3-28$ have equal magnitudes of $10.0 \mathrm{~m}$ and the angles are $\theta_{1}=30^{\circ}$ and $\theta_{2}=105^{\circ} .$ Find the
(a) $x$ and (b) $y$ components of their vector sum $\vec{r},(\mathrm{c})$ the magnitude of $\vec{r}$ and (d) the angle $\vec{r}$ makes with the positive direction of the $x$ axis.

Molika So

University of North Florida

Problem 16

For the displacement vectors $\vec{a}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}+(4.0 \mathrm{~m}) \hat{\mathrm{j}}$ and $\vec{b}=$
$(5.0 \mathrm{~m}) \hat{\mathrm{i}}+(-2.0 \mathrm{~m}) \hat{\mathrm{j}}$, give $\vec{a}+\vec{b}$ in
(a) unit-vector notation, and as (b) a magnitude and (c) an angle (relative to $\hat{1}$ ). Now give $\vec{b}-\vec{a}$ in (d) unit-vector notation, and as (e) a magnitude and (f) an angle. -

Molika So

University of North Florida

Problem 17

Three vectors $\vec{a}, \vec{b}$, and $\vec{c}$ each have a magnitude of $50 \mathrm{~m}$ and lie in an $x y$ plane. Their directions relative to the positive direction of the $x$ axis are $30^{\circ}, 195^{\circ}$, and $315^{\circ}$, respectively. What are
(a) the magnitude and (b) the angle of the vector $\vec{a}+\vec{b}+\vec{c}$, and
(c) the magnitude and (d) the angle of $\vec{a}-\vec{b}+\vec{c}$ ? What are the
(e) magnitude and (f) angle of a fourth vector $\vec{d}$ such that $(\vec{a}+\vec{b})-(\vec{c}+\vec{d})=0 ?$

KR

K Jagadeeswara Reddy

Numerade Educator

Problem 18

In the sum $\vec{A}+\vec{B}=\vec{C}$, vector $\vec{A}$ has a magnitude of $12.0 \mathrm{~m}$ and is angled $40.0^{\circ}$ counterclockwise from the $+x$ direction, and vector $\vec{C}$ has a magnitude of $15.0 \mathrm{~m}$ and is angled $20.0^{\circ}$ counterclockwise from the $-x$ direction. What are (a) the magnitude and (b) the angle (relative to $+x$ ) of $\vec{B}$ ?

Molika So

University of North Florida

Problem 19

In a game of lawn chess, where pieces are moved between the centers of squares that are each $1.00 \mathrm{~m}$ on edge, a knight is moved in the following way: (1) two squares forward, one square rightward; (2) two squares leftward, one square forward; (3) two squares forward, one square leftward. What are (a) the magnitude and (b) the angle (relative to "forward") of the knight's overall displacement for the series of three moves?

Zachary Warner

Numerade Educator

Problem 20

An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for $5.6 \mathrm{~km}$, but when the snow clears, he discovers that he actually traveled $7.8 \mathrm{~km}$ at $50^{\circ}$ north of due east. (a) How far and (b) in what direction must he now travel to reach base camp?

Keshav Singh

Numerade Educator

Problem 21

An ant, crazed by the Sun on a hot Texas afternoon, darts over an $x y$ plane scratched in the dirt. The $x$ and $y$ components of four consecutive darts are the following, all in centimeters: $(30.0,$, 40.0), $\left(b_{x},-70.0\right),\left(-20.0, c_{y}\right),(-80.0,-70.0)$. The overall displacement of the four darts has the $x y$ components $(-140,-20.0)$. What are (a) $b_{x}$ and (b) $c_{y} ?$ What are the (c) magnitude and (d) angle (relative to the positive direction of the $x$ axis) of the overall displacement?

Zachary Warner

Numerade Educator

Problem 22

(a) What is the sum of the following four vectors in unitvector notation? For that sum, what are (b) the magnitude, (c) the angle in degrees, and (d) the angle in radians?
$$\begin{array}{ll}\vec{E}: 6.00 \mathrm{~m} \mathrm{at}+0.900 \mathrm{rad} & \vec{F}: 5.00 \mathrm{~m} \text { at }-75.0^{\circ} \\\vec{G}: 4.00 \mathrm{~m} \mathrm{at}+1.20 \mathrm{rad} & \vec{H}: 6.00 \mathrm{~m} \text { at }-210^{\circ}\end{array}$$

Keshav Singh

Numerade Educator

Problem 23

If $\vec{B}$ is added to $\vec{C}=3.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}$, the result is a vector in the positive direction of the $y$ axis, with a magnitude equal to that of $\vec{C}$. What is the magnitude of $\vec{B}$ ?

Zachary Warner

Numerade Educator

Problem 24

Vector $\vec{A}$, which is directed along an $x$ axis, is to be added to vector $\vec{B}$, which has a magnitude of $7.0 \mathrm{~m}$. The sum is a third vector that is directed along the $y$ axis, with a magnitude that is $3.0$ times that of $\vec{A}$. What is that magnitude of $\vec{A}$ ?

Derek Walkama

Numerade Educator

Problem 25

Oasis $B$ is $25 \mathrm{~km}$ due east of oasis $A$. Starting from oasis
$A$, a camel walks $24 \mathrm{~km}$ in a direction $15^{\circ}$ south of east and then walks $8.0 \mathrm{~km}$ due north. How far is the camel then from oasis $B$ ?

Zachary Warner

Numerade Educator

Problem 26

What is the sum of the following four vectors in (a) unitvector notation, and as (b) a magnitude and (c) an angle?
$$\begin{array}{ll}\vec{A}=(2.00 \mathrm{~m}) \hat{\mathrm{i}}+(3.00 \mathrm{~m}) \hat{\mathrm{j}} & \vec{B}: 4.00 \mathrm{~m}, \text { at }+65.0^{\circ} \\
\vec{C}=(-4.00 \mathrm{~m}) \hat{\mathrm{i}}+(-6.00 \mathrm{~m}) \hat{\mathrm{j}} & \vec{D}: 5.00 \mathrm{~m}, \text { at }-235^{\circ}\end{array}$$

Keshav Singh

Numerade Educator

Problem 27

If $\vec{d}_{1}+\vec{d}_{2}=5 \vec{d}_{3}, \vec{d}_{1}-\vec{d}_{2}=3 \vec{d}_{3}$, and $\vec{d}_{3}=2 \hat{i}+4 \hat{j}$, then
what are, in unit-vector notation, (a) $\vec{d}_{1}$ and (b) $\vec{d}_{2}$ ?

Zachary Warner

Numerade Educator

Problem 28

Two beetles run across flat sand, starting at the same point. Beetle 1 runs $0.50 \mathrm{~m}$ due east, then $0.80 \mathrm{~m}$ at $30^{\circ}$ north of due east. Beetle 2 also makes two runs; the first is $1.6 \mathrm{~m}$ at $40^{\circ}$ east of due north. What must be (a) the magnitude and (b) the direction of its second run if it is to end up at the new location of beetle $1 ?$

Keshav Singh

Numerade Educator

Problem 29

Typical backyard ants often create a network of chemical trails for guidance. Extending outward from the nest, a trail branches (bifurcates) repeatedly, with $60^{\circ}$ between the branches. If a roaming ant chances upon a trail, it can tell the way to the nest at any branch point: If it is moving away from the nest, it has two choices of path requiring a small turn in its travel direction, either $30^{\circ}$ leftward or $30^{\circ}$ rightward. If it is moving toward the nest, it has only one such choice. Figure 3-29 shows a typical ant trail, with lettered straight sections of $2.0 \mathrm{~cm}$ length and symmetric bifurcation of $60^{\circ} .$ Path $v$ is parallel to the $y$ axis. What are the (a) magnitude and (b) angle (relative to the positive direction of the superimposed $x$ axis) ofan ant's displacement from the nest (find it in the figure) if the ant enters the trail at point $A$ ? What are the (c) magnitude and
(d) angle if it enters at point $B$ ?

Keshav Singh

Numerade Educator

Problem 30

Here are two vectors:
$$\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}-(3.0 \mathrm{~m}) \hat{\mathrm{j}} \text { and } \vec{b}=(6.0 \mathrm{~m}) \hat{\mathrm{i}}+(8.0 \mathrm{~m}) \hat{\mathrm{j}}$$
What are (a) the magnitude and (b) the angle (relative to $\hat{1}$ ) of $\vec{a}$ ? What are (c) the magnitude and (d) the angle of $\vec{b}$ ? What are (e) the magnitude and (f) the angle of $\vec{a}+\vec{b} ;(\mathrm{g})$ the magnitude and
(h) the angle of $\vec{b}-\vec{a} ;$ and (i) the magnitude and (j) the angle of $\vec{a}-\vec{b} ?(\mathrm{k})$ What is the angle between the directions of $\vec{b}-\vec{a}$ and $\vec{a}-\vec{b} ?$

Keshav Singh

Numerade Educator

Problem 31

In Fig. $3-30$, a vector $\vec{a}$ with a magnitude of $17.0 \mathrm{~m}$ is directed at angle $\theta=56.0^{\circ}$ counterclockwise from the $+x$ axis. What are the components (a) $u_{x}$ and (b) $u_{y}$ of the vector? A second coordinate system is inclined by angle $\theta^{\prime}=18.0^{\circ}$ with respect to the first. What are the components (c) $a_{x}^{\prime}$ and (d) $a_{y}^{\prime}$ in this primed coordinate system?

Keshav Singh

Numerade Educator

Problem 32

In Fig. 3-31, a cube of edge length $a$ sits with one corner at the origin of an $x y z$ coordinate system. $A$ body diagonal is a line that extends from one corner to another through the center. In unit-vector notation, what is the body diagonal that extends from the corner at (a) coordinates $(0,$, 0,0 ), (b) coordinates $(a, 0,0)$, (c) coordinates $(0, a, 0)$, and $(\mathrm{d})$ coordinates $(a, a, 0) ?(\mathrm{e})$ Determine the- angles that the body diagonals make with the adjacent edges.
(f) Determine the length of the body diagonals in terms of $a$.

Keshav Singh

Numerade Educator

Problem 33

For the vectors in Fig. $3-32$, with $a=4, b=3$, and $c=5$, what are (a) the magnitude and (b) the direction of $\vec{a} \times \vec{b}$, (c) the magnitude and (d) the direction of $\vec{a} \times \vec{c}$, and (e) the magnitude and (f) the direction of $\vec{b} \times \vec{c}$ ? (The $z$ axis is not shown.)

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 34

Two vectors are presented as $\vec{a}=3.0 \hat{\mathrm{i}}+5.0 \hat{\mathrm{j}}$ and $\vec{b}=2.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}$. Find
(a) $\vec{a} \times \vec{b}$, (b) $\vec{a} \cdot \vec{b}$, (c) $(\vec{a}+\vec{b}) \cdot \vec{b}$, and
(d) the component of $\vec{a}$ along the direction of $\vec{b}$. (Hint: For (d), consider Eq. $3-20$ and Fig. 3-18.)

Derek Walkama

Numerade Educator

Problem 35

Two vectors, $\vec{r}$ and $\vec{s}$, lie in the $x y$ plane. Their magnitudes are $4.50$ and $7.30$ units, respectively, and their directions are $320^{\circ}$ and $85.0^{\circ}$, respectively, as measured counterclockwise from the positive $x$ axis. What are the values of (a) $\vec{r} \cdot \vec{s}$ and (b) $\vec{r} \times \vec{s}$ ?

Zachary Warner

Numerade Educator

Problem 36

If $\vec{d}_{1}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $\vec{d}_{2}=-5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$, then what is
$\left(\vec{d}_{1}+\vec{d}_{2}\right) \cdot\left(\vec{d}_{1} \times 4 \vec{d}_{2}\right) ?$

Derek Walkama

Numerade Educator

Problem 37

Three vectors are given by $\vec{a}=3.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}-2.0 \hat{\mathrm{k}}$,
$\vec{b}=-1.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}$, and $\vec{c}=2.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}+1.0 \hat{\mathrm{k}}$. Find (a)
$\vec{a} \cdot(\vec{b} \times \vec{c})$, (b) $\vec{a} \cdot(\vec{b}+\vec{c})$, and $(\mathrm{c}) \vec{a} \times(\vec{b}+\vec{c})$

Keshav Singh

Numerade Educator

Problem 38

For the following three vectors, what is $3 \vec{C} \cdot(2 \vec{A} \times \vec{B}) ?$
$$\begin{aligned}
&\vec{A}=2.00 \hat{\mathrm{i}}+3.00 \hat{\mathrm{j}}-4.00 \hat{\mathrm{k}} \\
&\vec{B}=-3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}}+2.00 \hat{\mathrm{k}} \quad \vec{C}=7.00 \hat{\mathrm{i}}-8.00 \hat{\mathrm{j}}\end{aligned}$$

Derek Walkama

Numerade Educator

Problem 39

Vector $\vec{A}$ has a magnitude of $6.00$ units, vector $\vec{B}$ has a mag. nitude of $7.00$ units, and $\vec{A} \cdot \vec{B}$ has a value of $14.0$. What is the angle between the directions of $\vec{A}$ and $\vec{B}$ ?

Keshav Singh

Numerade Educator

Problem 40

Displacement $\vec{d}_{1}$ is in the $y z$ plane $63.0^{\circ}$ from the positive direction of the $y$ axis, has a positive $z$ component, and has a magnitude of $4.50 \mathrm{~m}$. Displacement $\vec{d}_{2}$ is in the $x z$ plane $30.0^{\circ}$ from the positive direction of the $x$ axis, has a positive $z$ component, and has magnitude $1.40 \mathrm{~m}$. What are (a) $\vec{d}_{1} \cdot \vec{d}_{2}$, (b) $\vec{d}_{1} \times \vec{d}_{2}$, and (c) the angle between $\vec{d}_{1}$ and $\vec{d}_{2}$ ?

Derek Walkama

Numerade Educator

Problem 41

Use the definition of scalar product, $\vec{a} \cdot \vec{b}=a b \cos \theta$, and the fact that $\vec{a} \cdot \vec{b}=a_{x} b_{x}+a_{y} b_{y}+a_{z} b_{z}$ to cal-
culate the angle between the two vectors given by $\vec{a}=3.0 \hat{\mathrm{i}}+$ $3.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$ and $\vec{b}=2.0 \hat{\mathrm{i}}+1.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$

Keshav Singh

Numerade Educator

Problem 42

In a meeting of mimes, mime 1 goes through a displacement $\vec{d}_{1}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}+(5.0 \mathrm{~m}) \hat{\mathrm{j}}$ and mime 2 goes through a displacement
$\vec{d}_{2}=(-3.0 \mathrm{~m}) \hat{\mathrm{i}}+(4.0 \mathrm{~m}) \hat{\mathrm{j}} .$ What are
(a) $\vec{d}_{1} \times \vec{d}_{2}$
(b) $\vec{d}_{1} \cdot \vec{d}_{2}$
(c) $\left(\vec{d}_{1}+\vec{d}_{2}\right) \cdot \vec{d}_{2}$, and
(d) the component of $\vec{d}_{1}$ along the direction of $\vec{d}_{2} ?$ (Hint: For (d), see Eq. 3-20 and Fig. $3-18 .)$

Derek Walkama

Numerade Educator

Problem 43

The three vectors in Fig. $3-33$ have magnitudes $a=3.00 \mathrm{~m}$, $b=4.00 \mathrm{~m}$, and $c=10.0 \mathrm{~m}$ and angle $\theta=30.0^{\circ} .$ What are (a) the $x$ component and (b) the $y$ component of $\vec{a} ;$ (c) the $x$ component and $(\mathrm{d})$ the $y$ com-ponent of $\vec{b}$; and (e) the $x$ component and (f) the $y$ component of $\vec{c}$ ? If $\vec{c}=p \vec{a}+q \vec{b}$, what are the values of $(\mathrm{g}) p$ and $(\mathrm{h}) q$ ?

Keshav Singh

Numerade Educator

Problem 44

In the product $\vec{F}=q \vec{v} \times \vec{B}$, take $q=2$,
$$\vec{v}=2.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}} \text { and } \vec{F}=4.0 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}$$
What then is $\vec{B}$ in unit-vector notation if $B_{x}=B_{y} ?$

Derek Walkama

Numerade Educator

Problem 45

Vectors $\vec{A}$ and $\vec{B}$ lie in an $x y$ plane. $\vec{A}$ has magnitude $8.00$ and angle $130^{\circ} ; \vec{B}$ has components $B_{x}=-7.72$ and $B_{y}=-9.20 .$ (a) What is $5 \vec{A} \cdot \vec{B} ?$ What is $4 \vec{A} \times 3 \vec{B}$ in (b) unit-vector notation and
(c) magnitude-angle notation with spherical coordinates (see Fig. 3-34)? (d) What is the angle between the directions of $\vec{A}$ and $4 \vec{A} \times 3 \vec{B} ?$ (Hint: Think a bit before you resort to a calculation.) What is $\vec{A}+3.00 \hat{\mathrm{k}}$ in (e) unit-vector notation and (f) magnitudeangle notation with spherical coordinates?

Keshav Singh

Numerade Educator

Problem 47

Vector $\vec{a}$ has a magnitude of $5.0 \mathrm{~m}$ and is directed east. Vector $\vec{b}$ has a magnitude of $4.0 \mathrm{~m}$ and is directed $35^{\circ}$ west of due north. What are (a) the magnitude and (b) the direction of $\vec{a}+\vec{b}$ ? What are (c) the magnitude and (d) the direction of $\vec{b}-\vec{a} ?$ (e) Draw a vector diagram for each combination.

Derek Walkama

Numerade Educator

Problem 47

Vectors $\vec{A}$ and $\vec{B}$ lie in an $x y$ plane. $\vec{A}$ has magnitude $8.00$ and angle $130^{\circ} ; \vec{B}$ has components $B_{x}=-7.72$ and $B_{y}=-9.20$. What are the angles between the negative direction of the $y$ axis and (a) the direction of $\vec{A}$, (b) the direction of the product $\vec{A} \times \vec{B}$, and $(\mathrm{c})$ the direction of $\vec{A} \times(\vec{B}+3.00 \hat{\mathrm{k}}) ?$

Zachary Warner

Numerade Educator

Problem 48

Two vectors $\vec{a}$ and $\vec{b}$ have the components, in meters, $a_{x}=3.2, a_{y}=1.6, b_{x}=0.50, b_{y}=4.5 .$ (a) Find the angle between the directions of $\vec{a}$ and $\vec{b}$. There are two vectors in the $x y$ plane that are perpendicular to $\vec{a}$ and have a magnitude of $5.0 \mathrm{~m} .$ One, vector $\vec{c}$, has a positive $x$ component and the other, vector $\vec{d}$, a negative $x$ component. What are (b) the $x$ component and (c) the $y$ component of vector $\vec{c}$, and (d) the $x$ component and (e) the $y$ component of vector $\vec{d}$ ?

Derek Walkama

Numerade Educator

Problem 49

A sailboat sets out from the U.S. side of Lake Erie for a point on the Canadian side, $90.0 \mathrm{~km}$ due north. The sailor, however, ends up $50.0 \mathrm{~km}$ due east of the starting point. (a) How far and (b) in what direction must the sailor now sail to reach the original destination?

Keshav Singh

Numerade Educator

Problem 50

Vector $\vec{d}_{1}$ is in the negative direction of a $y$ axis, and vector $\vec{d}_{2}$ is in the positive direction of an $x$ axis. What are the directions of
(a) $\vec{d}_{2} / 4$ and (b) $\vec{d}_{1} /(-4) ?$ What are the magnitudes of products (c) $\vec{d}_{1} \cdot \vec{d}_{2}$ and (d) $\vec{d}_{1} \cdot\left(\vec{d}_{2} / 4\right) ?$ What is the direction of the vector resulting from (e) $\vec{d}_{1} \times \vec{d}_{2}$ and (f) $\vec{d}_{2} \times \vec{d}_{1}$ ? What is the magnitude of the vector product in (g) part (e) and (h) part (f)? What are the (i) magnitude and (j) direction of $\vec{d}_{1} \times\left(\vec{d}_{2} / 4\right)$ ?

Derek Walkama

Numerade Educator

Problem 51

Rock faults are ruptures along which opposite faces of rock have slid past each other. In Fig. $3-35$, points $A$ and $B$ coincided before the rock in the foreground slid down to the right. The net displacement $\overrightarrow{A B}$ is along the plane of the fault. The horizontal component of $\overrightarrow{A B}$ is the strike-slip $A C$. The component of $\overrightarrow{A B}$ that is directed down the plane of the fault is the dip-slip $A D$. (a) What is the magnitude of the net displacement $\overrightarrow{A B}$ if the strike-slip is $22.0 \mathrm{~m}$ and the dip-slip is $17.0 \mathrm{~m} ?$ (b) If the plane of the fault is inclined at angle $\phi=52.0^{\circ}$ to the horizontal, what is the vertical component of $\overrightarrow{A B}$ ?

Keshav Singh

Numerade Educator

Problem 52

Here are three displacements, each measured in meters:
$\vec{d}_{1}=4.0 \hat{\mathrm{i}}+5.0 \hat{\mathrm{j}}-6.0 \hat{\mathrm{k}}, \quad \vec{d}_{2}=-1.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$, and $\vec{d}_{3}=$
$4.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}$. (a) What is $\vec{r}=\vec{d}_{1}-\vec{d}_{2}+\vec{d}_{3} ?$ (b) What is the
angle between $\vec{r}$ and the positive $z$ axis? (c) What is the component of $\vec{d}_{1}$ along the direction of $\vec{d}_{2} ?$ (d) What is the component of $\vec{d}_{1}$ that is perpendicular to the direction of $\vec{d}_{2}$ and in the plane of $\vec{d}_{1}$ and $\vec{d}_{2}$ ? (Hint: For (c), consider Eq. $3-20$ and Fig. $3-18 ;$ for $(\mathrm{d})$, consider Eq. $3-24 .$ )

Keshav Singh

Numerade Educator

Problem 53

A vector $\vec{a}$ of magnitude 10 units and another vector $\vec{b}$ of magnitude $6.0$ units differ in directions by $60^{\circ} .$ Find (a) the scalar product of the two vectors and (b) the magnitude of the vector product $\vec{a} \times \vec{b}$

Keshav Singh

Numerade Educator

Problem 54

For the vectors in Fig. 3 - 32 , with $a=4, b=3$, and $c=5$, calculate (a) $\vec{a} \cdot \vec{b}$, (b) $\vec{a} \cdot \vec{c}$, and $(\mathrm{c}) \vec{b} \cdot \vec{c} .$

Keshav Singh

Numerade Educator

Problem 55

A particle undergoes three successive displacements in plane, as follows: $\vec{d}_{1}, 4.00 \mathrm{~m}$ southwest; then $\vec{d}_{2}, 5.00 \mathrm{~m}$ east; an finally $\vec{d}_{3}, 6.00 \mathrm{~m}$ in a direction $60.0^{\circ}$ north of east. Choose a coon dinate system with the $y$ axis pointing north and the $x$ axis pointin east. What are (a) the $x$ component and (b) the $y$ component of $\vec{d}_{1}$ What are (c) the $x$ component and
(d) the $y$ component of $\vec{d}_{2}$ What are (e) the $x$ component and (f) the $y$ component of $\vec{d}_{3}$ Next, consider the net displacement of the particle for the thre successive displacements. What are (g) the $x$ component, (h) the component, (i) the magnitude, and (j) the direction of the net dis placement? If the particle is to return directly to the starting poin
(k) how far and (1) in what direction should it move?

Keshav Singh

Numerade Educator

Problem 56

Find the sum of the following four vectors in (a) unit-vecto notation, and as (b) a magnitude and (c) an angle relative to $+x$.
$\vec{P}: 10.0 \mathrm{~m}$, at $25.0^{\circ}$ counterclockwise from $+x$
Q: $12.0 \mathrm{~m}$, at $10.0^{\circ}$ counterclockwise from $+y$ $\vec{R}: 8.00 \mathrm{~m}$, at $20.0^{\circ}$ clockwise from $-y$
$\vec{S}: 9.00 \mathrm{~m}$, at $40.0^{\circ}$ counterclockwise from $-y$

Keshav Singh

Numerade Educator

Problem 57

If $\vec{B}$ is added to $\vec{A}$, the result is $6.0 \hat{\mathrm{i}}+1.0 \hat{\mathrm{j}} .$ If $\vec{B}$ is subtracted from $\vec{A}$, the result is $-4.0 \hat{\mathrm{i}}+7.0 \hat{\mathrm{j}}$. What is the magnitude of $\vec{A} ?$

Keshav Singh

Numerade Educator

Problem 58

A vector $\vec{d}$ has a magnitude of $2.5 \mathrm{~m}$ and points north. What are (a) the magnitude and (b) the direction of $4.0 \vec{d}$ ? What are (c) the magnitude and (d) the direction of $-3.0 \vec{d}$ ?

Derek Walkama

Numerade Educator

Problem 59

A has the magnitude $12.0 \mathrm{~m}$ and is angled $60.0^{\circ}$ counterclockwise from the positive direction of the $x$ axis of an $x y$ coordinate system. Also, $\vec{B}=(12.0 \mathrm{~m}) \hat{\mathrm{i}}+(8.00 \mathrm{~m}) \hat{\mathrm{j}}$ on that same coordinate
system. We now rotate the system counterclockwise about the origin by $20.0^{\circ}$ to form an $x^{\prime} y^{\prime}$ system. On this new system, what are (a) $\vec{A}$ and (b) $\vec{B}$, both in unit-vector notation?

Keshav Singh

Numerade Educator

Problem 60

If $\vec{a}-\vec{b}=2 \vec{c}, \vec{a}+\vec{b}=4 \vec{c}$, and $\vec{c}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}$, then what are
(a) $\vec{a}$ and $(\mathrm{b}) \vec{b} ?$

Derek Walkama

Numerade Educator

Problem 61

(a) In unit-vector notation, what is $\vec{r}=\vec{a}-\vec{b}+\vec{c}$ if $\vec{a}=5.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}-6.0 \hat{\mathrm{k}}, \vec{b}=-2.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$, and $\vec{c}=4.0 \hat{\mathrm{i}}+$
$3.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}} ?(\mathrm{~b})$ Calculate the angle between $\vec{r}$ and the positive $z$ axis. (c) What is the component of $\vec{a}$ along the direction of $\vec{b} ?$ (d) What is the component of $\vec{a}$ perpendicular to the direction of $\vec{b}$ but in the plane of $\vec{a}$ and $\vec{b}$ ? (Hint: For (c), see Eq. 3-20 and Fig. 3-18; for $(\mathrm{d})$, see Eq. $3-24 .$ )

Zachary Warner

Numerade Educator

Problem 62

A golfer takes three putts to get the ball into the hole. The first putt displaces the ball $3.66 \mathrm{~m}$ north, the second $1.83 \mathrm{~m}$ southeast, and the third $0.91 \mathrm{~m}$ southwest. What are (a) the magnitude and (b) the direction of the displacement needed to get the ball into the hole on the first putt?

Derek Walkama

Numerade Educator

Problem 63

Here are three vectors in meters:
$$\begin{aligned}&\vec{d}_{1}=-3.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}} \\
&\vec{d}_{2}=-2.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}} \\
&\vec{d}_{3}=2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}+1.0 \hat{\mathrm{k}}
\end{aligned}$$
What results from (a) $\vec{d}_{1} \cdot\left(\vec{d}_{2}+\vec{d}_{3}\right)$, (b) $\vec{d}_{1} \cdot\left(d_{2} \times \vec{d}_{3}\right)$, and
(c) $\vec{d}_{1} \times\left(\vec{d}_{2}+\vec{d}_{3}\right) ?$

Zachary Warner

Numerade Educator

Problem 64

A room has dimensions $3.00 \mathrm{~m}$ (height) $x$ $3.70 \mathrm{~m} \times 4.30 \mathrm{~m}$. A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) Could the length of its path be less than this magnitude? (c) Greater? (d) Equal? (e) Choose a suitable coordinate system and express the components of the displacement vector in that system in unit-vector notation. (f) If the fly walks, what is the length of the shortest path? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 65

A protester carries his sign of protest, starting from the origin of an $x y z$ coordinate system, with the $x y$ plane horizontal. He moves $40 \mathrm{~m}$ in the negative direction of the $x$ axis, then $20 \mathrm{~m}$ along a perpendicular path to his left, and then $25 \mathrm{~m}$ up a water tower. (a) In unit-vector notation, what is the displacement of the sign from start to end? (b) The sign then falls to the foot of the tower. What is the magnitude of the displacement of the sign from start to this new end?

Zachary Warner

Numerade Educator

Problem 66

Consider $\vec{a}$ in the positive direction of $x, \vec{b}$ in the positive direction of $y$, and a scalar $d$. What is the direction of $\vec{b} / d$ if $d$ is
(a) positive and (b) negative? What is the magnitude of (c) $\vec{a} \cdot \vec{b}$ and (d) $\vec{a} \cdot \vec{b} / d ?$ What is the direction of the vector resulting from
(e) $\vec{a} \times \vec{b}$ and (f) $\vec{b} \times \vec{a}$ ? (g) What is the magnitude of the vector product in (e)? (h) What is the magnitude of the vector product in
(f)? What are (i) the magnitude and (j) the direction of $\vec{a} \times \vec{b} / d$ if $d$ is positive?

Keshav Singh

Numerade Educator

Problem 67

Let $\hat{i}$ be directed to the east, $\hat{\mathrm{j}}$ be directed to the north, and $\hat{\mathrm{k}}$ be directed upward. What are the values of products (a) $\hat{i} \cdot \hat{k},(b)$ $(-\hat{\mathrm{k}}) \cdot(-\hat{\mathrm{j}})$, and $(\mathrm{c}) \hat{\mathrm{j}} \cdot(-\hat{\mathrm{j}}) ?$ What are the directions (such as east
or down of products $(\mathrm{d}) \hat{\mathrm{k}} \times \hat{\mathrm{j}},(\mathrm{e})(-\hat{\mathrm{i}}) \times(-\hat{\mathrm{j}})$, and $(\mathrm{f})(-\hat{\mathrm{k}}) \times(-\hat{\mathrm{j}}) ?$

Keshav Singh

Numerade Educator

Problem 68

A bank in downtown Boston is robbed (see the map in Fig. 3-36). To elude police, the robbers escape by helicopter, making three successive flights described by the following displacements: $32 \mathrm{~km}, 45^{\circ}$ south of east; $53 \mathrm{~km}, 26^{\circ}$ north of west; $26 \mathrm{~km}, 18^{\circ}$ east of south. At the end of the third flight they are captured. In what town are they apprehended?

Keshav Singh

Numerade Educator

Problem 69

A wheel with a radius of $45.0 \mathrm{~cm}$ rolls without slipping along a horizontal floor (Fig. 3-37). At time $t_{1}$, the dot $P$ painted on the rim of the wheel is at the point of contact between the wheel and the floor. At $\underline{a}$ later time $t_{2}$, the wheel has rolled through one-half of a revolution. What are (a) the magnitude and (b) the angle (relative to the floor) of the displacement of $P$ ?

Zachary Warner

Numerade Educator

Problem 70

A woman walks $250 \mathrm{~m}$ in the direction $30^{\circ}$ east of north, then $175 \mathrm{~m}$ directly east. Find (a) the magnitude and (b) the angle of her final displacement from the starting point. (c) Find the distance she walks. (d) Which is greater, that distance or the magnitude of her displacement?

Keshav Singh

Numerade Educator

Problem 71

A vector $\vec{d}$ has a magnitude $3.0 \mathrm{~m}$ and is directed south. What are (a) the magnitude and (b) the direction of the vector $5.0 \vec{d}$ ? What are (c) the magnitude and (d) the direction of the vector $-2.0 \vec{d}$ ?

Zachary Warner

Numerade Educator

Problem 72

A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: $\vec{d}_{1}$ for $0.40 \mathrm{~m}$ southwest (that is, at $45^{\circ}$ from directly south and from directly west), $\vec{d}_{2}$ for $0.50 \mathrm{~m}$ due east, $\vec{d}_{3}$ for $0.60 \mathrm{~m}$ at $60^{\circ}$ north of east. Let the positive $x$ direction be east and the positive $y$ direction be north. What are (a) the $x$ component and (b) the $y$ component of $\vec{d}_{1} ?$ Next, what are (c) the $x$ component and (d) the $y$ component of $\vec{d}_{2}$ ? Also, what are (e) the $x$ component and (f) the $y$ component of $\vec{d}_{3}$ ?

Keshav Singh

Numerade Educator

Problem 73

What are (g) the $x$ component, (h) the $y$ component, (i) the magnitude, and (j) the direction of the ant's net displacement? If the ant is to return directly to the starting point, $(\mathrm{k})$ how $\mathrm{far}$ and $(1)$ in what direction should it move? 173 Two vectors are given by $\vec{a}=3.0 \hat{\mathrm{i}}+5.0 \hat{\mathrm{j}}$ and $\vec{b}=2.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}$.
Find (a) $\vec{a} \times \vec{b},($ b) $\vec{a} \cdot \vec{b}$, (c) $(\vec{a}+\vec{b}) \cdot \vec{b}$, and $(\mathrm{d})$ the component of $\vec{a}$ along the direction of $\vec{b}$.

Keshav Singh

Numerade Educator

Problem 74

Vector $\vec{a}$ lies in the $y z$ plane $63.0^{\circ}$ from the positive direction of the $y$ axis, has a positive $z$ component, and has magnitude $3.20$ units. Vector $\vec{b}$ lies in the $x z$ plane $48.0^{\circ}$ from the positive direction of the $x$ axis, has a positive $z$ component, and has magnitude $1.40$ units. Find (a) $\vec{a} \cdot \vec{b},($ b) $\vec{a} \times \vec{b}$, and (c) the angle between $\vec{a}$ and $\vec{b}$.

Derek Walkama

Numerade Educator

Problem 75

Find (a) "north cross west," (b) "down dot south," (c) "east cross up," (d) "west dot west," and (e) "south cross south." Let each "vector" have unit magnitude. 76 A vector $\vec{B}$, with a magnitude of $8.0 \mathrm{~m}$, is added to a vector $\vec{A}$, which lies along an $x$ axis. The sum of these two vectors is a third vector that lies along the $y$ axis and has a magnitude that is twice the magnitude of $\vec{A}$. What is the magnitude of $\vec{A}$ ?

Keshav Singh

Numerade Educator

Problem 76

A vector $\vec{B}$, with a magnitude of $8.0 \mathrm{~m}$, is added to a vector $\vec{A}$, which lies along an $x$ axis. The sum of these two vectors is a third vector that lies along the $y$ axis and has a magnitude that is twice the magnitude of $\vec{A}$. What is the magnitude of $\vec{A}$ ?

Derek Walkama

Numerade Educator

Problem 77

A man goes for a walk, starting from the origin of an $x y z$ coordinate system, with the $x y$ plane horizontal and the $x$ axis eastward. Carrying a bad penny, he walks $1300 \mathrm{~m}$ east, $2200 \mathrm{~m}$ north, and then drops the penny from a cliff $410 \mathrm{~m}$ high. (a) In unit-vector notation, what is the displacement of the penny from start to its landing point? (b) When the man returns to the origin, what is the magnitude of his displacement for the return trip?

Keshav Singh

Numerade Educator

Problem 78

What is the magnitude of $\vec{a} \times(\vec{b} \times \vec{a})$ if $a=3.90, b=2.70$, and the angle between the two vectors is $63.0^{\circ}$ ?

Derek Walkama

Numerade Educator

Problem 79

In Fig. 3-38, the magnitude of $\vec{a}$ is $4.3$, the magnitude of $\vec{b}$ is 5.4, and $\phi=46^{\circ}$. Find the area of the triangle contained between the two vectors and the thin diagonal line.

Zachary Warner

Numerade Educator