Chapter 3 Skill Building

MasteringPhysics: Assignment Print View

Vector Addition and Subtraction

In general it is best to conceptualize vectors as arrows in space, and then to make calculations with them using their components. (You must first specify a coordinate system in order to find the components of each arrow.) This problem gives you some practice with the components.

Let vectors , , and . Calculate the following, and express

your answers as ordered triplets of values separated by commas.

Part A

ANSWER: = 3.00,−5.00,

−4.00

Part B

ANSWER: = −5.00,4.00,0

Part C

ANSWER: = −6.00,4.00,3.00

Part D

ANSWER: = −3.00,−2.00,

−11.0

Part E

ANSWER: = −11.0,14.0,8.00

Part F

ANSWER: = 17.0,−12.0,

−6.00

http://session.masteringphysics.com/myct/assignmentPrint?assignmentID=1040719 (1 of 21)4/11/2006 2:42:30 PM


Vector Addition: Geometry and Components

Learning Goal: To understand how vectors may be added using geometry or by representing them with components.

Fundamentally, vectors are quantities that possess both magnitude and direction. In physics problems, it is best to think of vectors as arrows, and usually it is best to manipulate them using components. In this problem we consider the addition of two vectors using both of these methods. We will emphasize that one method is easier to conceptualize and the other is more suited to manipulations.

Consider adding the vectors and , which have lengths and , respectively, and where

makes an angle from the direction of .

In vector notation the sum is represented by

.

Part A

Which of the following procedures will

add the vectors and ?

ANSWER: Put the tail of on

the arrow of ; goes

from the tail of to the

arrow of

Put the tail of on

the tail of ; goes

from the arrow of to

the arrow of

Put the tail of on



the tail of ; goes

from the arrow of to

the arrow of

Calculate the magnitude as the sum of the lengths and the direction as midway

between and .

It is equally valid to put the tail of on

the arrow of ; then goes from the

tail of to the arrow of .

Part B

Find , the length of , the sum of and .

Hint B.1

Hint not displayed

Part B.2

Part not displayed

Express in terms of , , and angle

.

ANSWER:

=

Part C



Find the angle that the vector makes with vector .

Hint C.1

Hint not displayed

Express in terms of and any of the quantities given in the problem introduction ( , , and/

or ) as well as any necessary constants. Use asin for arcsine

ANSWER: =

Part D

To manipulate these vectors using vector components, we must first choose a coordinate system. In this case choosing means specifying the angle of the x axis. The y axis must be perpendicular to

this and by convention is oriented radians counterclockwise from the x axis.

Indicate whether the following statement is true or false:

There is only one unique coordinate system in which vector components can be added.

ANSWER: true false

Part E

The key point is that you are completely free to choose any coordinate system you want in which to manipulate the vectors. It is a matter of convenience only, and so you must consider which orientation will simplify finding the components of the given vectors and interpreting the results in that coordinate system to get the required answer. Considering these factors, and knowing that you

are going to be required to find the length of and its angle with respect to , which of the

following orientations simplifies the calculation the most?



ANSWER: The x axis should be oriented along

The x axis should be oriented along

The x axis should be oriented along

Part F

Find the components of in the coordinate system shown.

Express your answer as an ordered pair: x component, y component ; in terms of and .

ANSWER:

=

Part G

In the same coordinate system, what are the components of ?

Express your answer as an ordered pair separated by a comma. Give your answer in terms of variables defined in the introduction ( , , and ).

ANSWER: =

This should show you how easy it is to add vectors using components. Subtraction is similar except that the components must be subtracted rather than added, and this makes it important to

know whether you are finding or . (Note that .)



Although adding vectors using components is clearly the easier path, you probably have no immediate picture in your mind to go along with this procedure. Conversely, you probably think of adding vectors in the way we've drawn the figure for Part B. This justifies the following maxim: Think about vectors geometrically; add vectors using components.

Components of Vectors

Shown is a 10 by 10 grid, with coordinate axes x and y . The grid runs from -5 to 5 on both axes. Drawn on this grid are four vectors,

labeled through . This problem will

ask you various questions about these vectors. All answers should be in decimal notation, unless otherwise specified.

Part A

What is the x component of ?

Hint A.1

Hint not displayed

Express your answer to two significant figures.

ANSWER: = 2.50

Part B

What is the y component of ?

Express your answer to the nearest integer.

ANSWER: = 3.00

Part C



What is the y component of ?

Hint C.1

Hint not displayed


ANSWER: = −3.00

Part D

What is the component of ?

Hint D.1

Hint not displayed


ANSWER: = −2.00

The following questions will ask you to give both components of vectors using the ordered pairs method. In this method, the x component is written first, followed by a comma, and then the y

component. For example, the components of would be written 2.5,3 in ordered pair notation.

The answers below are all integers, so estimate the components to the nearest whole number.

Part E

In ordered pair notation, write down the components of vector .

Express your answers to the nearest integer.

ANSWER: , =

2.00,−3.00

Part F

In ordered pair notation, write down the components of vector .

Express your answers to the nearest integer.

ANSWER: , =

2.00,−3.00

Part G



What is true about and ? Choose from the pulldown list below.

ANSWER:

Resolving Vector Components with Trigonometry

Often a vector is specified by a magnitude and a direction; for example, a rope with

tension exerts a force of magnitude

in a direction 35 north of east. This is a

good way to think of vectors; however, to calculate results with vectors, it is best to select a coordinate system and manipulate the components of the vectors in that coordinate system.

Part A

Find the components of the vector

with length = 1.00 and angle =10.0

with respect to the x axis as shown.

Hint A.1

Hint not displayed

Write the components (numerically) in the form .

ANSWER: = 0.985,0.174

Part B



Find the components of the vector with length = 1.00 and angle =20.0 with respect to the

x axis as shown.

Hint B.1

Hint not displayed

Write the components (numerically) in the form .

ANSWER: = 0.940,0.342

The components of still have the same form, that is, , despite 's

placement with respect to the y axis on the drawing.

Part C

Find the components of the vector with length = 1.00 and angle 30.0 as shown.

Part C.1

Part not displayed

Hint C.2

Hint not displayed

Write the components in the form .

ANSWER: = −0.500,0.866

Frames of Reference and Relative Velocity

Learning Goal: To understand the concept of frames of reference and relative velocity.

Any event can be considered from different perspectives. In particular, the description of the motion of a given object would depend on the motion of the person describing the motion, the observer. Any such observer, equipped with a coordinate system and a stopwatch, forms a frame of reference.

Usually, we assume that a frame of reference is attached to some object. For instance, "the frame of reference of earth" means the observer is at rest relative to earth.

In this problem, you will practice viewing motion and calculating its parameters in different frames of reference. The rules of vector addition and subtraction will be useful here. To denote the velocities in different frames of reference, we will use the following notation:



means "the velocity of object P in the frame of reference (or with respect to, or

relative to) object Q."

Consider four cars: A, B, C, and D. The cars are moving along a straight stretch of a road; their velocities are shown in the diagram.

The first several questions refer to this diagram.

Part A

Consider the statement of the problem: "...[the cars'] velocities are shown in the diagram." It is reasonable to assume that the velocities on the diagram are given relative to which of the following?

ANSWER: car E (not shown in the diagram)

earth

the sun

For almost all problems where the events described obviously take place on earth, the default frame of reference is earth, and the words "relative to earth" are often omitted from descriptions of motion.

Part B

What is ?

ANSWER: 12 to the right

12 to the left

0

Since car B is not moving with respect to earth, is simply the velocity of A relative to earth,

which is already shown in the diagram!

Part C



What is ?


12 to the left

0

To the driver of car A, car B appears to be moving to the left and approaching car A at 12 .

Recall that, if you are driving in a car, street signs seem to be coming toward you at the same speed that you appear to be moving with respect to earth. Notice that "toward you" is opposite to the direction of your motion with respect to earth.

Part D

Can you determine the distance between A and B?

ANSWER: yes

no

The information about the velocities tells us nothing about the distances between the objects (or, for that matter, about their accelerations). Additional information is needed to answer this question.

Part E

What is ?


3 to the left

27 to the right

27 to the left

To the driver of car C, car A appears to be moving to the right and approaching car C at 27 .

In general,

or

.

Part F



What is ?


5 to the left

35 to the right

35 to the left

Part G

What is ?


5 to the left

35 to the right

35 to the left

In general, .

Part H

Car E, also moving along the same stretch of road, is not shown in the diagram. It is known that,

from the point of view of driver A, car E is moving at 8 . What is the speed of car E relative to

car D?

Hint H.1

Hint not displayed

ANSWER: 0

12

20

24

28

40

either 24 or 40

either 12 or 28



either 20 or 40

Part I

Car G, also moving along the same stretch of road, is not shown in the diagram. The following information is given about the speed of car G:

.

What is the speed ?

Express your answer in meters per second.

ANSWER: = 2.00

Of course, the speed that you just found is the speed relative to earth. Let us now try a slightly more challenging question.

Part J

Car H, also moving along the same stretch of road, is not shown in the diagram. The following information is given about the speed of car H:

.

What is ?


ANSWER: = 26.0

Part K

Refer to the two previous parts to find .


ANSWER: = 28.0

You may have noticed that cars G and H have the same speed with respect to car A. However, they move in opposite directions, which is why the relative speed of these two cars is not zero.



Let us now consider a two-dimensional case. This time, there are several cars--A, B, C, and D--moving as shown on the diagram . All velocities are given in the frame of reference of earth.

In answering the following questions, you may use

,

where "E" means earth. It also helps to imagine yourself as the driver of one of the cars.

Part L

Which of the vectors best represents ?

ANSWER: 1

2

3

4

Part M

Which of the vectors best represents ?



ANSWER: 1

2

3

4

Part N

Find .

Express your answer in meters per second to three significant figures.

ANSWER: = 22.4

Vector Components--Review

Learning Goal: To introduce you to vectors and the use of sine and cosine for a triangle when resolving components.Vectors are an important part of the language of science, mathematics, and engineering. They are used to discuss multivariable calculus, electrical circuits with oscillating currents, stress and strain in structures and materials, and flows of atmospheres and fluids, and they have many other applications. Resolving a vector into components is a precursor to computing things with or about a vector quantity. Because position, velocity, acceleration, force, momentum, and angular momentum are all vector quantities, resolving vectors into components is the most important skill required in a mechanics course.

The figure shows the components of , and , along the x and y axes of the coordinate system,

respectively. The components of a vector depend on the coordinate system's orientation, the key being the angle between the vector and the coordinate axes, often designated . Part A



The figure shows the standard way of measuring the angle. is measured to the vector from the x

axis, and counterclockwise is positive.

Express and in terms of the length of the

vector and the angle , with the components

separated by a comma.

ANSWER: , =

In principle, you can determine the components of

any vector with these expressions. If lies in one

of the other quadrants of the plane, will be an

angle larger than 90 degrees (or in radians)

and and will have the appropriate

signs and values.

Unfortunately this way of representing , though

mathematically correct, leads to equations that must be simplified using trig identities such as

and

.

These must be used to reduce all trig functions present in your equations to

either or . Unless you perform this

followup step flawlessly, you will fail to recoginze that

,

and your equations will not simplify so that you can progress further toward a solution. Therefore,



it is best to express all components in terms of

either or , with

between 0 and 90 degrees (or 0 and in radians),

and determine the signs of the trig functions by knowing in which quadrant the vector lies.

Part B

When you resolve a vector into components, the components must have the

form or . The signs depend on which quadrant the vector lies in, and there

will be one component with and the other with .

In real problems the optimal coordinate system is often rotated so that the x axis is not horizontal. Furthermore, most vectors will not lie in the first quadrant. To assign the sine and cosine correctly for vectors at arbitrary angles, you must figure out which angle is and then properly reorient the

definitional triangle.

As an example, consider the vector shown in the diagram labeled "tilted axes," where you

know the angle between and the y axis.

Which of the various ways of orienting the definitional triangle must be used to

resolve into components in the tilted

coordinate system shown? (In the figures, the hypotenuse is orange, the side adjacent to is red, and the side opposite

is yellow.)

Hint B.1

Hint not displayed

Hint B.2

Hint not displayed



Indicate the number of the figure with the correct orientation.

ANSWER: 1 2 3 4

Part C



Choose the correct procedure for determining the components of a vector in a given coordinate system from this list:

ANSWER: Align the adjacent side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle.

Align the hypotenuse of a right triangle with the vector and an adjacent side along a coordinate direction with as the included angle.

Align the opposite side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle.

Align the hypotenuse of a right triangle with the vector and the opposite side along a coordinate direction with as the included angle.

Part D

The space around a coordinate system is conventionally divided into four numbered quadrants depending on the signs of the x and y coordinates. Consider the following conditions:

A. ,

B. ,

C. ,

D. ,

Which of these conditions are true in which quadrants?

Write the answer in the following way: If A were true in the third quadrant, B in the second, C in the first, and D in the fourth, enter "3, 2, 1, 4" as your response.



ANSWER: 1,4,2,3

Part E

Now find the components and of in the tilted coordinate system of Part B.

Express your answer in terms of the length of the vector and the angle , with the components

separated by a comma.

ANSWER: , =

A Canoe on a River

A canoe has a velocity of 0.500 southeast relative to the earth. The canoe is on a river that is

flowing at 0.530 east relative to the earth.

Part A

Find the magnitude of the velocity of

the canoe relative to the river.

Hint A.1

Hint not displayed

Part A.2

Part not displayed

Part A.3

Part not displayed


ANSWER: = 0.395

Part B



Find the direction of the velocity of the canoe relative to the river.

Hint B.1

Hint not displayed

Part B.2

Part not displayed

Express your answer as an angle measured south of west.

ANSWER: 63.5 degrees south of west


Chapter 3 Skill Building

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