I
The Project Physics Course
Programmed Instruction
Vectors 1 The Concept of Vectors
Vectors 2 Adding 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
Copyright (?) 1974, Project Physic*
Copyright (^C^ 1971. Project Physics
All Rights Reserved
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
start with these examples.
Vectors 2 Adding Vectors
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
you can already answer these frames, you need
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
3. Always write your answer before you look at the given answer.
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
Answer Space
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
/
Answer Space
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.
Answer to A
(i) a number (with or without units)
(ii) number (with or without units)
and a direction.
Answer to A
(i)
X * Y
(ii) 3.7 units
Answer A
i
7
F
y
/!
/
L
F
(
Answer Space
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
Answer Space
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
Answer to B
(i) T, 7. and P^
(ii) m and 6
Answer to B
Resultant
Resultant Force, F„ shown.
An;
(i)
(ii) 45 "^ below Korizontol,
50 m sec.
If your answers to the sample
questions were correct, the
remainder of the program is
optionol.
Sample Question C
Answer Space
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
Answer Space
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.
Answer to C
Your answer is correct only if the
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.
Answer to C
/^
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
about 1700 meters, measuring
center to center
A1
Vector quantities add
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
must add to give D
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
answer the following questions:
(i) What building will you reach if you start at the Washington Monu-
ment and travel 2600 meters due east?
(ii) What was your displacement?
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
resultant of adding ^aq and
(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
on this railroad hopper car.
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
about 15 units
(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