Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 4

Vectors - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Alfjad Alfjad

Alfjad Alfjad

Numerade Educator

Problem 2

A bank in downtown Boston is robbed (see the map in Fig. 4-18. 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? (Use the geometrical method to add these displacements on the map.)

Donald Albin

Numerade Educator

Problem 3

The motion of three objects is shown in the motion diagrams $(a),(b)$, and $(c)$ of Fig. $4-19 .$ In each case the object is shown at three equally spaced times. A circle with no arrow indicates a velocity of zero magnitude, such as the final velocity in diagram $(a) .$ Indicate for each part which number is next to the arrow on the right side of the diagram that best shows the direction of the change in velocity. (Hint: Use the techniques developed in Section 4-3 to draw vectors representing the difference in velocity in each case.) Note: This exercise is adapted from a conceptual exercise developed by Dennis Albers of Columbia College.

Donald Albin

Numerade Educator

Problem 4

A pea leaves a pea shooter at a speed of $5.4 \mathrm{~m} / \mathrm{s}$. It makes an angle of $+30^{\circ}$ with respect to the horizontal.
(a) Calculate the $x$ -component of the pea's initial velocity.
(b) Calculate the $y$ -component of the pea's initial velocity.
(c) Write an expression for the pea's velocity, $\vec{v}$, using unit vectors for the $x$ direction and the $y$ direction.

Alfjad Alfjad

Numerade Educator

Problem 5

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 $250^{\circ}$ counterclockwise from the positive direction of the $x$ axis and its magnitude is $7.3 \mathrm{~m}$ ?

Alfjad Alfjad

Numerade Educator

Problem 6

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

Derek Walkama

Numerade Educator

Problem 7

The $x$ -component of vector $A$ 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 8

A displacement vector $\vec{r}$ in the $x y$ plane is $15 \mathrm{~m}$ long and directed as shown in Fig. 4-20. Determine
(a) the $x$ -component and
(b) the $y$ -component of the vector.

Alfjad Alfjad

Numerade Educator

Problem 9

A wheel with a radius of $45.0 \mathrm{~cm}$ rolls without slipping along a horizontal floor (Fig. 4-21). 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 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 $P$ during this interval?

Alfjad Alfjad

Numerade Educator

Problem 10

Rock faults are ruptures along which opposite faces of rock have slid past each other. In Fig. 4-22 points $A$ and $B$ coincided before the rock in the foreground slid down to the right. The net displacement $A B$ 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 directly 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 $52.0^{\circ}$ to the horizontal, what is the vertical component of $\overrightarrow{A B}$ ?

Donald Albin

Numerade Educator

Problem 11

A room has dimensions $3.00 \mathrm{~m}$ (height) $\times 3.70 \mathrm{~m} \times$ $4.30 \mathrm{~m}$. 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 than this magnitude? (d) Equal to this magnitude? (e) Choose a suitable coordinate system and find the components of the displacement vector in that system. (f) If the fly walks rather than flies, what is the length of the shortest path it can take? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

Alfjad Alfjad

Numerade Educator

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 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?

Alfjad Alfjad

Numerade Educator

Problem 14

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?

Derek Walkama

Numerade Educator

Problem 15

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

Alfjad Alfjad

Numerade Educator

Problem 16

Find the (a) $x$ - (b) $y$ - and (c) $z$ -components of the sum $\Delta \vec{r}$ of the displacements $\Delta \vec{c}$ and $\Delta \vec{d}$ whose components in meters along the three axes are $\Delta c_{x}=7.4, \Delta c_{y}=-3.8$ $\Delta c_{z}=-6.1 ; \Delta d_{x}=4.4, \Delta d_{y}=-2.0, \Delta d_{z}=3.3$

Alfjad Alfjad

Numerade Educator

Problem 17

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 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.

Alfjad Alfjad

Numerade Educator

Problem 18

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

Alfjad Alfjad

Numerade Educator

Problem 19

Two vectors are given by
$$\begin{array}{l}
\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}-(3.0 \mathrm{~m}) \hat{\mathrm{j}}+(1.0 \mathrm{~m}) \hat{\mathrm{k}} \\
\vec{b}=(-1.0 \mathrm{~m}) \hat{\mathrm{i}}+(1.0 \mathrm{~m}) \hat{\mathrm{j}}+(4.0 \mathrm{~m}) \hat{\mathrm{k}} .
\end{array}$$
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 20

Three 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 $\hat{\mathrm{i}}$ ) 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 $\vec{a}-\vec{b}$ ?
(k) What is the angle between the directions of $\vec{b}-\vec{a}$ and $\vec{a}-\vec{b}$ ?

Alfjad Alfjad

Numerade Educator

Problem 21

Three vectors $\vec{a}$, and $\vec{b}$, 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 $(\mathrm{c})$ the magnitude and $(\mathrm{d})$ the angle of $\vec{a}-\vec{b}+\vec{c} ?$ What are (e) the magnitude and (f) the angle of a fourth vector $\vec{d}$ such that $(\vec{a}+\vec{b})-(\vec{c}+\vec{d})=0$

Alfjad Alfjad

Numerade Educator

Problem 22

What is the sum of the following four vectors in
(a) unit-vector notation and (b) magnitude-angle notation? For the latter, give the angle in both degrees and radians. Positive angles are counterclockwise from the positive direction of the $x$ axis; negative angles are clockwise.
$$
\begin{array}{ll}
\vec{E}: 6.00 \mathrm{~m} \text { at }+0.900 \mathrm{rad} & \vec{F}: 5.00 \mathrm{~m} \text { at }-75.0^{\circ} \\
\vec{G}: 4.00 \mathrm{~m} \text { at }+1.20 \mathrm{rad} & \vec{H}: 6.00 \mathrm{~m} \text { at }-210^{\circ}
\end{array}
$$

Alfjad Alfjad

Numerade Educator

Problem 23

The two vectors $\vec{a}$ and $\vec{b}$ in Fig. 4-23 have equal magnitudes of $10.0 \mathrm{~m}$. Find (a) the $x$ -component and (b) the $y$ -component of their vector sum $\vec{r},(\mathrm{c})$ the magnitude of $\vec{r}$, and $(\mathrm{d})$ the angle $\vec{r}$ makes with the positive direction of the $x$ axis.

Alfjad Alfjad

Numerade Educator

Problem 24

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 25

Prove that two vectors must have equal magnitudes if their sum is perpendicular to their difference.

Alfjad Alfjad

Numerade Educator

Problem 26

Find the sum of the following four vectors in
(a) unit-vector 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 $+x$
$\vec{Q}: 12.0 \mathrm{~m}$, 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 $-y$

Alfjad Alfjad

Numerade Educator

Problem 27

Two vectors of magnitudes $a$ and $b$ make an angle $\theta$ with each other when placed tail to tail. Prove, by taking components along two perpendicular axes, that
$$r=\sqrt{a^{2}+b^{2}+2 a b \cos \theta}$$
gives the magnitude of the sum $\vec{r}$ of the two vectors.

Alfjad Alfjad

Numerade Educator

Problem 28

What is the sum of the following four vectors in (a) unit-vector notation, and as (b) a magnitude and (c) an angle? Positive angles are counterclockwise from the positive direction of the $x$ axis; negative angles are clockwise.
$$
\begin{array}{ll}
\vec{A}=(2.00 \mathrm{mi}) \hat{\mathrm{i}}+(3.00 \mathrm{mi}) \hat{\mathrm{j}} & \vec{B}: 4.00 \mathrm{~m}, \text { at }+65.0^{\circ} \\
\vec{C}=(-4.00 \mathrm{~m}) \hat{\mathrm{i}}-(6.00 \mathrm{~m}) \hat{\mathrm{j}} & \vec{D}: 5.00 \mathrm{~m}, \text { at }-235^{\circ}
\end{array}
$$

Alfjad Alfjad

Numerade Educator

Problem 29

(a) Using unit vectors, write expressions for the four body diagonals (the straight lines from one corner to another through the center) of a cube in terms of its edges, which have length $a$. (b) Determine the angles that the body diagonals make with the adjacent edges. (c) Determine the length of the body diagonals in terms of $a$.

Alfjad Alfjad

Numerade Educator

Problem 30

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 31

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 $\vec{A}$ ?

Alfjad Alfjad

Numerade Educator

Problem 32

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

Alfjad Alfjad

Numerade Educator

Problem 33

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?

Nishant Kumar

Numerade Educator

Problem 34

The three vectors in Fig. 4-24 have magnitudes $a=3.00 \mathrm{~m}, b=4.00 \mathrm{~m}$, and $c=10.0 \mathrm{~m}$. What are (a) the $x$ -component and (b) the $y$ -component of $\vec{a} ;$ (c) the $x$ -component and (d) the $y$ -component of $\vec{b}$; and $(\mathrm{e})$ 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 $(\mathrm{h}) q ?$

Alfjad Alfjad

Numerade Educator

Problem 35

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 36

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 37

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?

Alfjad Alfjad

Numerade Educator

Problem 38

In each case below, sketch the velocity vector. Find the magnitude and direction of motion with respect to the $x$ axis of the coordinate system:
(a) $\vec{v}=(2.45 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(3.67 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$
(b) $\vec{v}=(-2.45 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(5.20 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$

Alfjad Alfjad

Numerade Educator

Problem 39

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 40

A fire ant, searching for hot sauce in a picnic area, no 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 (that is, directly east), $\vec{d}_{3}$ for $0.60 \mathrm{~m}$ at $60^{\circ}$ north of east (that is $60.0^{\circ}$ toward the north from due 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 $\vec{d}_{1} ?$ 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 $\vec{d}_{3}$ ? What are $(\mathrm{g})$ 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 far and (I) in what direction should it move?

Alfjad Alfjad

Numerade Educator

Problem 41

A heavy piece of machinery is raised by sliding it $12.5 \mathrm{~m}$ along a plank oriented at $20.0^{\circ}$ to the horizontal, as shown in Fig. $4-25 .$ (a) How high above its original position is it raised? (b) How far is it moved horizontally?

Alfjad Alfjad

Numerade Educator

Problem 42

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 $1 ?$

Alfjad Alfjad

Numerade Educator

Problem 43

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 $-70$ ), 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 44

The magnitude and angle of $\vec{A}$, which lies in an $x y$ plane, are $4.00$ and $130^{\circ}$, respectively. What are the components (a) $A_{x}$ and (b) $A_{y}$ ? Vector $\vec{B}$ also lies in the $x y$ plane, and it has components $B_{x}=-3.86$ and $B_{y}=-4.60 .$ What is $\vec{A}+\vec{B}$ in (c) magnitude-angle notation and (d) unit-vector notation? In (e) unit-vector notation and (f) magnitude-angle notation, find $\vec{C}$ such that $\vec{A}-\vec{C}$ $=\vec{B}$. (g) Which of the vector diagrams in Fig. 4-26 correctly show the relationship between those three vectors?

Alfjad Alfjad

Numerade Educator

Problem 45

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 46

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 47

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 48

A protester carries his sign of protest $40 \mathrm{~m}$ along a straight path, then $20 \mathrm{~m}$ along a perpendicular path to his left, and then $25 \mathrm{~m}$ up a water tower. (a) Choose and describe a coordinate system for this motion. In terms of that system and in unitvector 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?

Alfjad Alfjad

Numerade Educator

Problem 49

In Fig. $4.27$, a vector $\vec{a}$ with a magnitude of $17.0 \mathrm{~m}$ is directed $56.0^{\circ}$ counterclockwise from the $+x$ axis, as shown. What are the components (a) $a_{x}$ and (b) $a_{y}$ of the vector? A second coordinate system is inclined by $18.0^{\circ}$ with respect to the first. What are the components (c) $a_{x}^{\prime}$ and (d) $a_{y}^{\prime}$ in this primed coordinate system?

Alfjad Alfjad

Numerade Educator

Problem 50

Consider how the components of a vector in the plane change if I change the reference point. Suppose I start with a coordinate system with an origin at $O$. An arbitrary vector $\vec{r}=x \hat{\mathrm{i}}+y \hat{\mathrm{j}}$ with coordinates $(x, y)$ specifies a point in this system. Suppose also that I have another point $O^{\prime}$ specified in this coordinate system by a vector $\vec{A}=A_{x} \hat{\mathrm{i}}+A_{y} \hat{\mathrm{j}}$. If I change my origin to $O^{\prime}$ (without rotating the axes), what would the coordinates be for the point specified by $\vec{r}$ ?

Alfjad Alfjad

Numerade Educator

Problem 51

$\vec{A}$ 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?

Alfjad Alfjad

Numerade Educator