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The Project Physics Course

Programmed Instruction

Vectors 1 The Concept of Vectors

Vectors 3 Components of Vectors

INTRODUCTION

You are about to use a programmed text.
You should try to use this booklet where there
ore no distractions — a quiet classroom or a study
area at home, for instance. Do not hesitate to
seek help if you do not understand some problem.
Programmed texts require your active porticipa-
tion and ore designed to challenge you to some
degree. Their sole purpose is to teach, not to
quiz you.

This book is designed so that you can work
through one program at a time. The first program.
Vectors 1, runs page by page across the top of each
page. Vectors 2 parallels it, running through the mid-
dle part of each page, and Vectors 3 similarly across
the bottom.

This publication is one of the many instructional moterials
developed for tho Project Physics Course. These ma-
terials include Texts, Handbooks, Teacher Resource Books,
Readers, Programmed Instruction Booklets, Film Loops,
Transparencies, 16mm films and laboratory equipment.
Development of the course has profited from the help of
many colleagues listed in the text units.

Directors of Project Physics

Gerald Holton, Department of Physics, Harvard

University
F. James Rutherford, Chairman, Department of

Science Education, New York University
Fletcher G. Watson, Harvard Graduate School

of Education

ISBN 0-03-089642-8
012 OOK 987

Project Phyiics it a registered trademark

A Component of the
Protect Physics Course

Distributed by

Molt, Rinehart and Winston

New York — Toronto

Cover Art by Andrew Ahlgren

Vectors 1 The Concept of Vectors

You are familiar with signs such a s \^ONEJ^{^!!j

[SUBWAyJ that indicate a direction. You have also
seen signs which give a magnitude such as

OR

MAXIMUM
35 TONS
CAPACITY

This program is about quantities that hove both a
direction and a numerical value. These are called
vectors and they are very important in physics.

You are already familiar with some ex-
amples of vectors. This port of the program will

Adding vectors is on important technique
for you to understand and be able to use.
After going through this set of programmed
materials you will be able to add two or more
vectors together and obtain the resultant vec-
tor. The next three sample questions represent
the kinds of questions you should be able to
answer after you have finished Vectors 2. if
not take Vectors 2. 't that case you can go on
to Vecto--- 3.

Vectors 3 Components of Vectors

When we use a vector to represent a
physical situation, we may wish to find the
component of that vector in a given direction.
This is Part III of the series of programmed
instruction booklets on vectors. In this part,
you will learn how to separate vectors into
components and how to obtain a vector from
its components.

The two sample questions that follow
illustrate the objectives of this part of the
program. Vectors 3. If you find that you
can answer these two questions correctly,
you need not work through the program.

INSTRUCTIONS

1. Frames: Each frame contains a question. Answer the question by writing in the blank space next to the frame.

Frames ore numbered 1, 2, 3, . . .

2. Answer Blocks: To find an answer to a frame, turn the page. Answer blocks ore numbered Al, A2, A3, . . .

This booklet is designed so that you can compare your answer witfi the given answer by folding
back the page, like this:

1 -

1

U-

"75^,

2 ;
1

4. If you get the right answers to the sample questions, you do not hove to complete the program.

INSTRUCTIONS: Same as for Program 1, above.

INSTRUCTIONS: Same as for Program 1, o' we.

Sample Question A

Complete this sentence if you can:

A scalar quantity can be expressed by (i)

quantity must be expressed by both (ii)

, but a vector

Sample Question A

Given are two vectors, X and Y,
represented by the arrows drawn here.

(i) Draw an arrow to represent the vector
sum (resultant).

(ii) Give its magnitude

/

2 units

Sample Question A

An arrow is shown that
represents a force vector F.

(i) Draw Fy, the component
of F in the y-direction.

(ii) Draw Fx, the component
of F in the x-direction.

(i) a number (with or without units)

(ii) number (with or without units)
and a direction.

(i)

X * Y

(ii) 3.7 units

i

7

F

y

/!

/

L

F

(

Sample Question B

It is important to be able to distinguish between vector and scalar
quantities in equations.

(i) List all of the vector quantities in the equation

T= mT+ 6P!

(ii) List all of the scalar quantities in the same equation.

Sample Question B

Three forces acting on an object, 0, can be repre-
sented by arrows as drawn below. What is the resultant
force on the object, that is, what is the vector sum of
the three forces?

Sample Question B

Given Vx and Vy:

(i) Construct and draw v.

(ii) Give the direction and
magnitude of v.

scale: 50

(i) T, 7. and P^
(ii) m and 6

Resultant

Resultant Force, F„ shown.

An;

(i)

(ii) 45 "^ below Korizontol,
50 m sec.

questions were correct, the
remainder of the program is
optionol.

Sample Question C

Suppose the wind is blowing from the
northeast at 12 m/sec. Draw an arrow
that represents this wind velocity to
the scale given.

10

15 20
-I I

Sample Question C

Forces F,, F2 and F3 (from the last frame) are shown
acting on a car. You found the resultant force by adding these
vectors together tip-to-tail as shown at the left.

What should the magnitude of F, have been if you wanted
the resultant force to be zero?

Draw the vector B that must be added to A to give C.

arrow you draw points in the same
direction as this one and is the
same length.

If you answered all 3 sample questions
correctly, you are ready for the
Vectors 2 program.

If not, begin with question 1 on the
next poge.

/^

New F,

F, , F; ♦• F3 =

if the mognitude of F, is 3.5 units.

A1

Now turn the page to begin Vectors 1. Remember to proceed through
the book from left to right, confining your attention to the top frame
on each page.

Now turn the page to begin Vectors 2. Remember, left to right,
middle frames only.

Draw two perpendicular vectors that add to give F.

-U

JUDDCII

WASHINGTON, D. C :^g^jyj^vj|

ooMmvnoN lau.

5:^ sg D^^' Wi

n®^!K □ M] C^SPm^^^

LimJ

^&(

t-«MlJ

;r!-f

Scale

(meters)
500

■ I .

1000

V»cton Part I
^ Mop of CantTol Section o*
j — 1 ( — I I — ^ WoAifiBtoo, O.C.,U.S.A.

^gS§^

snr

The Parallelogram Law

A vector is an entity having both magnitude and direction; vectors also have the property of
addition by the parallelogram law as shown here, where A and B represent two vector quantities.

It can be
drawn either

The vector sum of A t B is C and can be drown in two ways. Both ways of drawing the parollelo-
grom low shown above are equivalent, but the "tip-to-tail" method on the right will be shown to-
the more powerful since it can be extended easily to more than two vectors.

There are many physical quantities which hove both direction and magnitude and odd to-
gether according to the parallelogram low. In Part I of the vectors program the displacement
vector was introduced, and Port II will begin with the addition of displacements.

A2

possible solutions:

NOTE; There are on infLnit*
number of solutions.

Questions 1 through 16 require the map of Washington, D.C.,
shown to the left.

Find the location of the Lincoln Memorial and the Jefferson Memo-
rial on the map of Washington, D.C. A straight line is shown be-
tween the memorials. According to the scale of the map, the dis-
tance between the Lincoln and Jefferson Memorials is

meters.

(Hint: One way to use the scale on the map is to copy it off the
edge of a piece of paper which can be placed along any line you
wish to measure.)

Read the panel on the opposite page.

You learned in Part I of the program that a vector quantity has
both magnitude and direction.

What other property will a vector quantity have?

Martha walked from the post office to the bus stop.

Her displacement is represented by the arrow marked D
on the map.

(i) How many blocks / Po^t y
east did she walk? \omceJ

(ii)How many blocks
south did she walk?

/ M

W

Oak St,

Elm St,

Park St.!

A1

center to center

A1

according to the paral-
lelogram low.

A3

(!) 6 blocks east
(ii) 2 blocks south

From the compass directions on the mop we can see that

the Jefferson Memorial is located 1700 meters of

the Lincoln Memorial.

Let us use vectors to represent a trip
around the city block. The first leg of
the trip starts at intersection A, and is
represented by dAo, the displacement

vector drawn from A to B.

(i) What is the magnitude of the vector

■^AB?
(ii) What is its direction?

Scale:

1 cm = 100 m

W-^

-^ E

The diagram below shows that A + B = C.

Two vectors which add to give a third vector are called
components of that vector.

In this example, (i)
are components of (iii).

and (i

A2

southeast

A2

(i) 250 meters (approx.)
(ii) north

A4

(i) A (or B)
(ii) B" (or A)
(iii)C

Locate the White House, and find the distance and direction
of the White House from the Jefferson Memorial.

On the panel draw the second leg of the
trip around the block, namely from B to C.

(i) Give the direction and magni tude of
the displacement vector dof--

(ii) Give the total distance traveled on
the first two legs of this trip.

The two paths marked / post ^
1 and 2 yield the same \officeJ

displacement vector D.

Also, the easterly and
southerly components

independently of the ^

path. ~ c5 ^ ^ -^

What is the magnitude of the southerly component of D?

A3

approximately 2100 meters to the north

A3

(i) a few degrees North of East
170 meters

(ii) 420 meters

(A to B = 250 m, B to C = 170 m)

A5

2 blocks

One of the important concepts of physics is that of displacement:
it is the straight line distance and direction between the initial and
final locations of an object. Use the map of Washington, D.C., to

(i) What building will you reach if you start at the Washington Monu-
ment and travel 2600 meters due east?

Draw the vector dip between
points A and C. (This goes diago-
nally across the block.)

(i) Give the magnitude and direc-
tion of d A p .

(ii) What is the difference (in
meters) between the distance
traveled from points A to B to
C, and the rrxignitude of the
vector d^C ?

The dashed line represents the actual path Martha took from
the post office to the bus stop. Her displacement D does not de-
pend on her path and the components of D likewise do not depend

on her path.

/^ — ^ I

/ post Y

What is the mag- '^ office^
nitude of the compo-
nent of D in the
easterly direction?

A4

(i) the U.S. capifol

(ii) 2600 m east from the
Washington Monument

A4

(i) 330 m

a few degrees North of NortKeost

(ii) 90 m difference

A6

6 blocks

(i) What would be your displacement if you traveled from the Capitol
to the White House?

(ii) What IS the dispiacement if something is moved from the White
House to the Washington Monument?

The displacement vector from A to
C, dxr / is the resultant of adding
d^g and dg^.

The displacement vector d^^p is the

(ii) What is the resultant of

'BC

jnd 6qq?

(iii) Draw the resultant of dgp
and df-Q on the diagram at
the right.

The vector F represents the force exerted by the rope on
the wagon. We can separate the force into vertical and horizontal
components.

(i) Draw the component of F in the vertical direction. Label it F^.
This component tends to lift the wagon.

(ii) Draw the component of F in the horizontal direction. Label it

• ^. This component of the force is responsible for the motion

of the wagon along the ground

A5

(i) 2900 m, approximately northwest
(octually 290' from north)

(ii) 1100 m south (octually slightly
east of south)

A5

CO dcD

(ii) dgD

A7

Tra^^te?^

A displacement can be represented ^Krj^^
by an arrow in a mop. The length of ^<i
the arrow represents a scale drawing »
of the actual displacement. ra^H^FI

IIBl

What displacement is shown?

^0

AVI ^5[

The final leg of the trip around the
block, from intersection D to A, is
given by the displacement vector

^DA-

Draw the vector sum of d/-Q and
^DA-

The arrow labeled Fgrav
represents the force of gravity

The component of Fgrav P®''"
pendicular to the track is
balanced by the opposite force
of the track on the wheels.

(i) Draw the component of

Fgrav that is perpendicular

to the track. Label it Fj^.

(ii) Draw the component of Fgrav that is parallel to the track.
Label it F„.

A6

White House to Washington Monument
(1100 m south)

A6

A8

grov

(i) Draw an arrow on the map to represent the displacement of a
person who has walked from the Washington Monument to the
Jefferson Memorial. (Hint: If you are not sure how to do this,
recall the definition of displacement in Frame 4.)

(ii) Draw a broken line on the map to show the shortest path for
walking on dry ground from the Washington Monument to the
Jefferson Memorial.

(iii) Is the path length the same as the displacement?

(iv) Does the choice of path change the displacement?

The four legs of the trip around the block can be represented by
the four separate vectors shown here.

'AB

^CD

What is the sum of these four vectors?

Here is an expanded diagram from Frame 8
The magnitude of Fg^^^ is 120,000 N.

(i) Find the magnitude of Fj_.
(ii) Find the magnitude of Fn .

scale: '■■■■'

50,000N

grav

(iii) no (it changes the
path length, but not the
displacement, which is
defined as the straight-
I ine distance.)

A7 i

S^^JIS

A7

A9

(i) 120,000 N
(ii) 30,000 N

On the map of Washington, D.C., there is on arrow which

indicated that the displacement of New York City from

Washington is •

distance? direction?

8

If the vector C is the sum of vectors A and B, we can write:

(i) Given A and B as shown,
draw the vector sum C.

(ii) Find the direction and
magnitude of C by
measuring the scale
drawing.

10 . L J t

In general, components of a vector are constructed as the sides ot
a parallelogram which has the vector as the diagonal. The angle betwee
the sides of the parallelogram may be any value; however, the physical
analysis is often easiest if this
is chosen to be 90°. The preced-
ing examples of the wagon and
the hopper car illustrate the use-
fulness of components that are
at right angles.

As an example of non-per-
pendicular components, take the

vector Fgrav from before and re-
solve it into components in the
q and r directions. Label the
components F and F^,. Be sure
to draw these components as vectors.

2 units
Scole: I 1 •

A8

320 Icm northeast

A8

(i^

(ii) direction: 43° from A.
magnitude: 5.? units.

A10

9 1

Note that the distance scale at the bottom of the map is for |
measurements inside Washington, and the displacement to more re- |
mote places such as New York City is represented with another |
scale. It is not essential that the arrow representing a displace- |
ment vector be drawn to the same scale as the map. |

Pittsburgh, Pennsylvania, is approximately 320 kilometers to the |
northwest of Washington. Draw an arrow by which you can repre- |
sent this displacement. 1

(Use the same scale as the arrow showing the displacement of 1
New York City.) 1

Two arrows representing the vectors S and T are drawn i

separately. S and T cannot be added without shifting them so |
that they touch. The most useful way to make this shift is so I
that the pointed "tip" of one touches the blunt "tail" of the 1
other. 1

(i) Redraw S and T with •
the tip of S touching j ■
the tail of T. . / t 1

/ !

(ii) Drew the vector sum / i
of S and T on the tip- 1
to-tail drawing. .

11 1

The previous frames have shown that a vector may
be resolved into components along any chosen axis. -

Now, given the components, it can be seen thot a ■

vector is the (vector) sum of its components. ■ ■*

1 b

1 y

Given B and B , find B. |

X

A9

A9

10

/

(ii)

All

10

Quantities that have both magnitude and direction ore called vectors.
Quantities that have a magnitude but no direction are called scalars.
Is the displacement shown below a scalar or a vector?

10

(!) Shift the arrow representing the vec-
tor Z so that its tail is touching the
tip of Q.

(ii) If R = Q + Z, draw an arrow repre-
senting R.

12

The ground exerts a pe-pendicular
force Fj. on the skier and the cable
pulling the skier exerts a force Fn .

The friction between the skis
and snow is negligible.

(i) Construct and draw the arrow repre-
senting the net force (Fnet) o^ ^^^
of the cable and the ground on the

(ii) What is the direction and magnitude
of the net force?

A10

vector

A10

(i)

*a

r^

00

'a

/

r^*

(ii) vertical (upward)
22 units of fore*

11

Quantities that have only a magnitude are called scalars.
Those quantities that have both magnitude and direction are called
vectors.

100

150

Is the position of the 50 meter mark on the scale a vector or a
scalar?

11

iTr F+ G^, Find h" by adding F
and G with the tip-to-tail method
in both of these ways:

(i) shifting F to the tip of G.

(ii) shifting G to the tip of F.

(iii) Do both procedures give
the same result?

13

The diagram shows a particle striking a barrier
and rebounding elastically.

(i) Resolve each of the velocity vectors into
components which are perpendicular to the
wall and parallel to the wall.

(ii) Which component of velocity did not change
during the interaction?

All

scalar

All

(i)

(ii)

(iii) Yes

A13

The component of velocity
parallel to the wall does
not change during the
interaction.

(ii) V

12

A scalar quantity can be expressed by a single number
(with or without units), but a vector must have both

12

The clear advantage of using the tip- to-tail method of graphically
adding vectors can be seen when three or more vectors are to be added
We have already seen this in the example of the city block. The ad-
dition is performed by
making a "chain" of
vectors. Then the sum
(or resultant) is found
by drawing the arrow
from the tail of the first
to the head of the lost
arrow in the chain.

Draw the resultant for U + P + S

14

Here is the same event
again.

Describe the change of the
component of velocity
peqtendicular to the wall.

A12

magnitude and direction

A12

^*P.c^

A14

The component of velocity
perpendicular to the wall
reverses direction but does
not change in magnitude.

13

Are the following pictures representations of vectors, of
scalers, or of neither?

(i)

(ii)

450 Miles

Son Froncisco to San Diego

To Chicago

13

(1) Redraw U, E and Y tip-to-tail.

(ii) Draw the vector sum of U + E + Y.

15

A ball has components of velocity
Vx and Vy as shown in the diagram.

(i) Construct and draw v.

(ii) Give the direction and
magnitude of v.

X

"^

N

)

V

\

X J

i

Y

scale: 50 m/sec

A13

(i) vector (a displacement)
(ii) neither (only direction)

A13

NOTE: As the reduced sketches below indi<
any sequence of V, E. ond Y will give the s
resultant.

/

/

/

/ /

A15

(i)

(ii) 45^" below horizontal.
50 m sec

14 1

On the map of Washington, D.C., there is an orrow representing |
the wind velocity. The arrow indicates that the wind is blowing from |
th«» (i) nt n <:p<>ed of (ii) |

14 j

(i) Redraw M, N and tip-to-tail. •
(ii) Draw the vector sum W, ^ ■

where M + N + = W. ~~^y^ '

You hove now completed all three programs in this book.
Understanding and being able to use vectors should be helpful
to you in many ways.

If ever you wish to refresh your memory on Vectors, you can
cover up the answer space with a sheet of blank paper and
quickly run through the frames again.

A14

(i) southeast

(ii) 9 m/sec (about 20 miles/hr)

A14

NOTE: Any sequence of M, N, and
will give the some W.

15 1

The speed and direction of the wind is a vector quantity, and .
therefore it can be represented by an arrow drawn to scale. Suppose ■
the wind changed and is now coming from the west at 18 m/sec. ■

Draw the new wind direction, and indicate the new wind speed by i
making the arrow of the proper length (using the other wind arrow i
as a guide). 1

15 j

/ 1

(i) Redraw the vectors J, K and L v j
tip-to-tail. \ j

(ii) T+ 1< + r = m! Draw the V^^^ ^ 1

arrow representing M. | |

J j

(iii) Does the order in which you '

redraw the vectors affect M ? T .

A15

wind speed = 18 rrv/\$ec

m

(This is twice as long as the length
shown for a wind speed of 9 rv sec.)

A15

(i)
(ii)

(iii)

1

16 1

To the same scale what is the length of the arrow needed to repre- '
sent a wind speed of 27 meters/sec?

16 [

Given M, , Mj, M3 as shown, and ^

M, ^ M2 + M3 = M4 ^ '"^Z 1

FindM^^. ^ ^^3 / 1

\ 1

A16

three times the length for 9m sec

A16

M4 is zero

17

Whenever we encounter a physical quantity — such as speed
force, energy, or whatever — it is useful for us to know whether or
not it involves direction. Those quantities that involve direction
as well OS magnitude ore called

(i)

(ii) Does the pull each team exerts on the rope in the tug-of-war
involve a direction?

17

If a", ^ A^ ♦ A3 = "5 and
A, and A2 are as shown, con-
struct the vector A3 that
satisfies this equation.

A17

(i) vectors
(ii)yes

A17

18

When we encounter a physical
quantity that is a scalar we mean it
has no

(i)

(ii) Is the diameter of the water wheel
shown here a vector or a scalar?

18

Force is a vector quantity. Each of
the cars shown here is exerting a force on
the large wooden box.

Below each car draw an arrow to indicate
the direction of the force each car exerts
on the object to which it is hitched.

A18

(i) direction
(ii) scolar

A18

1

19 1

Four boys ore shown pushing a |
car. The force each boy exerts on the |
car is a |

(i) qiinntity^ nnd the

number of boys pushing the car is a .

(ii) qiinntity.

19

Suppose the small cor (1'
force the other car (2) exerts.

1 2

pulls with half the i

m-H

II

v I

Draw arrows represe

nting the fo

ce each car exerts. 1

A19

(i) vector
(ii) scalar

A19

(I)

(2)

NOTE: These arrows con be of ony
length except thot (1) must be |ust
one-half the length of (2).

20 j

When writing one usually draws a small arrow over the symbol i
used for vector quantities. For example, in the equation i

F = m a, 1

F represents a vector quantity, the force, and a represents an ac- |
celeration in the same direction as F. The letter m represents a
a scalar, mass. 1
(i) List all vector quantities in the equation •

T = ma + 6N ■

(ii) List all of the scalar quantities in the same equotion. •

20 j

(i) What is the sum of the two pulls of the cars, namely the j
resultant force exerted on the box by both cars pulling

together? Assume the pulling forces; F, = 5 units (to the •

left)

F2 = 10 units (to the 1
left) 1

(ii) Draw the resultant force ( Fp ) ? I

A20

(i)T,^N

(Did yoo put the orrows over
the symbols?)

(ii) m, 6

A20

(i) 15 units of force to the left

(ii)

(1) (2)

Resolton*

%

21

The negative of a vector quantity is represented by an arrow
in the reverse direction. For example if A is represented by

X ., -r. . ,, -X

ifTis >/

^. then —A is represented by
draw — B.

21

Two cars are shown pulling on a
wooden box. The pulling force of each

car is represented by the vectors F,

and F2 (note the units).

(i) Construct the vector sum Fp of these
forces using the tip-to-tail method. (If
you are not sure how to do this, refer
to Frame II.)

(ii) What is the direction and magni-
tude of the sum Fp ?

(iii) Write an equation to represent
the relation between F) , Fj
and Fp .

A21

y

Did you draw - B to the proper
length? It is a vector in the
direction opposite to B but
having the same magnitude.

A21

(i)

(ii) to the left and a few degrees
below horizontal, magnitude

(iii)F,

F, = F.

22 1

If -C is \^ give a full label to: >^ '

22 [

Suppose that two cars were pulling •
an object, and that each is exerting a .
force represented by the arrowns shown ^ ■

'"'■ - - ''%^. 1

(i) Find the vector sum Fi + Fz- ^*"^v,.^ |
(ii) Draw an arrow representing a force \. ^/^ 1
vector F3 such that F, + F2 + F3 = 0. ^^^

(iii) If F3 is the force exerted on the ^9^

object by a third car, what is the ^W^ !
resultant force on the object? '

A22

(iii)

23 j

This ends Vectors 1. |

You have learned to distinguish between vectors and scalars. You |
have drawn vector quantities to scale, and you hove learned that a |
negative vector is in the opposite direction from the corresponding 1
positive vector. 1

You are now ready to learn to add vector quantities. See the pro- '
gram Vectors 2. It begins at the front of this book and occupies '
the middle of each page. •

23 1

Three forces acting on an object can be represented |
by arrows as drawn below. 1

Draw an arrow to represent the resultant force Fp |
on the object. 1

^ 'i A . !

(Hint: If you are not sure how to do this, refer to Frame 15.) ■

A23

24

1

Forces F,, Fj and F, (from the last frame) are sfiown

acting on object C. You found the resultant force Fp by

adding these three vectors together ""tip-to-tail" in Frame

23. What mcanitude ihouic F. hove tn order to make the

resuitcnt force zero?

i

• 1

, ^' A .

Force scale ^^^^

2 units

A24

3 units

25

Given are two vectors, X and Y, represented by

the arrows drawn here.

(i) Draw on arrow to represent the vector sum.

1

^

(ii) Give its magnitude. ^

y 1

/ \

2 units
1 t

A25

(i)

\$om X + Y

(ii) 3.7 units

I

This ends Vectors 2. I

You have learned how to add two or more vectors together and to I

draw the resultant vector. Also, given two vectors, you have •

practiced finding a third vector that would just balance the first I

two vectors so that the sum of the three was zero. I

If you would now like to learn about components of vectors, see ■

the program Vectors 3. It begins on the bottom part of the first ,

page of this book. ,

0-03-089642-8

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