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The Project Physics Course 

Programmed Instruction 

Vectors 1 The Concept of Vectors 

Vectors 2 Adding Vectors 

Vectors 3 Components of Vectors 


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 

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 


35 TONS 

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. 


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 - 




2 ; 

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 


X * Y 

(ii) 3.7 units 

Answer A 










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 Force, F„ shown. 



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

If your answers to the sample 
questions were correct, the 
remainder of the program is 

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. 


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. 


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. 



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

ooMmvnoN lau. 

5:^ sg D^^' Wi 

n®^!K □ M] C^SPm^^^ 







■ I . 


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



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. 


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 


(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 


Oak St, 

Elm St, 

Park St.! 


about 1700 meters, measuring 
center to center 


Vector quantities add 
according to the paral- 
lelogram low. 


(!) 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 

(ii) What is its direction? 


1 cm = 100 m 


-^ 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 




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


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

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? 


approximately 2100 meters to the north 


(i) a few degrees North of East 
170 meters 

(ii) 420 meters 

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


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? 


(i) the U.S. capifol 

(ii) 2600 m east from the 
Washington Monument 


(i) 330 m 

a few degrees North of NortKeost 

(ii) 90 m difference 


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 


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 

(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 


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

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


CO dcD 

(ii) dgD 



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 


What displacement is shown? 


AVI ^5[ 

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


Draw the vector sum of d/-Q and 

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„. 


White House to Washington Monument 
(1100 m south) 




(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. 



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: '■■■■' 



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

A7 i 




(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? 


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 

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 • 


320 Icm northeast 



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


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. | 









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? 


(!) 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. 


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? 











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


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



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


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? 


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? 






(iii) Yes 


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

(ii) V 


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


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 


Here is the same event 

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


magnitude and direction 




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


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



450 Miles 

Son Froncisco to San Diego 

To Chicago 


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

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


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 J 



scale: 50 m/sec 


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


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




/ / 



(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. 


(i) southeast 

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


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 . 


wind speed = 18 rrv/$ec 


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





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 


three times the length for 9m sec 


M4 is zero 


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 


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


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


(i) vectors 



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


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


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. 


(i) direction 
(ii) scolar 



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. 


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

1 2 

pulls with half the i 



v I 

Draw arrows represe 

nting the fo 

ce each car exerts. 1 


(i) vector 
(ii) scalar 




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 • 


F2 = 10 units (to the 1 
left) 1 

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



(Did yoo put the orrows over 
the symbols?) 

(ii) m, 6 


(i) 15 units of force to the left 


(1) (2) 




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. 


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 . 



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



(ii) to the left and a few degrees 
below horizontal, magnitude 
about 15 units 


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? ' 



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.) ■ 




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? 


• 1 

, ^' A . 

Force scale ^^^^ 

2 units 


3 units 


Given are two vectors, X and Y, represented by 

the arrows drawn here. 

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



(ii) Give its magnitude. ^ 

y 1 

/ \ 

2 units 
1 t 



$om X + Y 

(ii) 3.7 units 


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. ,