3-2 vectors and scalars is a number with units. it can be positive or negative. example: distance,...

1

3-2 Vectors and Scalars

Is a number with units. It can be positive or negative.

Example: distance, mass, speed, Temperature…

Chapter 3: VECTORS3-2 Vectors and Scalars

Scalar

Is a quantity with both direction and magnitude.

Example: velocity, force, displacement…

Vector

2


The figure shows a displacement vector where B

the particle undergoes a displacement from A to B. A B’

The displacement vector is represented by B” A’

an arrow pointing from A to B. (we will use A”

here triangle arrowheads for arrows that represent vectors)

In the figure, the arrows from A to B, from A’ to B’, and from A” to

B” have the same magnitude and direction. They represent identical

displacement vectors.

3


• The displacement vector tells us nothing about the actual

path that the particle takes. In the figure, for example, all

three paths connecting points A and B correspond to the

same displacement vector.

B A

4

Vector is written with an arrow over the symbol .

The magnitude of the vector is written as .

3-3 Adding Vectors Geometrically

ff

��

��=��+ ��

��

��tail

head

5


Two important properties:

1. Adding to gives the same result as adding to , that is,

(commutative law)

Start Finish

abba

ba

a

b

a

b

a

a

b

b


2. For more than two vectors, the addition can be done

in any order.

That is:

(associative law)

a

cb

)()( cbacba

)( cba

cba

)(

b

c

ba

cb

)( cba

cba

)(

cba

ba

c

Vector substraction :

The vector is a vector with the same magnitude as

but the opposite direction.


��

− ��

��

��− ��=��−��

7

0)( aa

8

Set up two dimensional coordinate system.

3-4 Components of vectors

is measured relative to the axis.

𝒙

𝒚

𝒐

��

𝒂𝒙

𝒂𝒚

Component)

Component)

figure) side (see 0 and 0 : For vector

change.not do components

itsdirection, its changinghout vector wit

a shiftyou if that shows figure above The

figure. above in the 0 and 0

axesy and x thealong

vector of components theare and

yx

yx

yx

bbb

aa

aaa

a

xa

ya

form a the right triangle shown in figure. The vector is completely determined by either: the pair ( ) or the pair ( ).It is possible to obtain any of these pairs in termsof the other pair using the right triangle as

follows:

and a

yx aaa and ,

a

yx aa and

11

Given the magnitude and direction of a vector , we get

its component and .

3-4 Components of vectors

Given the components and of a vector , we get its

magnitude and direction .

13

3-5 Unit vectors

A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction.

• The unit vectors in the positive directions of the x, y and z directions are labeled , and , respectively (see figure)

i

j

kx

y

z

i j k

For example, if a vector has the scalar component and

, we write it as:

3-5 Unit vectors

Component Component

a

iaxˆ

jayˆ

x

y

. ˆ and ˆ vectorscomponent theaddingby get vector We jaiaa yx

are called the vector components of . jaia yxˆ and ˆ a

• In two dimensions any vector like can be

written as: a

jaiaa yxˆˆ

15

If a vector is written as:

and

Then

3-6 Adding vectors by components

The component of

16

To Add vectors and by using components method

1. Calculate the scalar components: , , and . 2. Calculate the components of the sum : 3. Combine the components of to get itself:

For vector subtraction: and

a

b

xa yaxb

yb

r

, yyyxxx barbar r

r

jrirr yxˆˆ

)( babad

yyyxxx badbad and

jdidd yxˆˆ

19

Multiplying a Vector by a Scalar:

3-8 Multiplying Vectors

Multiplication of a vector by a scalar gives a vector .

o If : and have the same direction.

o If : and have the opposite direction.

The magnitude of = ( the magnitude of )x(absolute value of )

as

a

s

a

a

2a

2

20


The Scalar or Dot Product:

The scalar product of the vectors and is written as :

o : magnitude of

o : magnitude of

o : angle between and

�� . ��=𝒂𝒃𝒄𝒐𝒔∅

��

��∅

21

= the product of the magnitude of one of the vectors by the scalar component of the second vector along the direction of the first vector (see figure):

= the component of along the direction of

= the component of along the direction of

Notice that:

ba

.

)cos()cos(. bababa

cosb b

a

cosa a

b

abba

..

Component of along direction of is acos

��

��

𝜽

Component of along direction of is bcos

22

• In unit vector notation, we write:

• Each vector component of the first vector is to be dotted by each vector component of the second vector. For example:

and

• Where the angle between and is 0 and that between and is 90. By doing so, we get:

)ˆˆˆ).(ˆˆˆ(. kbjbibkajaiaba zyxzyx

xxxxxxxx babaiibaibia 0cos)1)(1()ˆ.ˆ(ˆ.ˆ

090cos)1)(1()ˆ.ˆ(ˆ.ˆ yxyxyx bajibajbia

zzyyxx babababa

.

i ii j

23

In terms of their components, the dot (scalar) product of two

vectors and is written as :

and

�� . ��=𝒂𝒙𝒃𝒙+𝒂𝒚 𝒃𝒚

Consider the case of two dimensional vectors and a

b

25

The Vector or Cross Product:


The vector product or cross product of the vectors and

is written as : and is another vector , where . The magnitude of is defined by

The vector is perpendicular to the plane of the two vectors

and

The direction of is given by the right hand rule. . . Here is the smaller angle between and a b

26

Right Hand Rule for Vector Product:

• To obtain the direction of vector , where we use the right-hand rule as follows (see figure on page 28):

1. Place the vectors and tail to tail

2. Imagine a line that is perpendicular to their plane where they meet.

3. Place your right hand around that line in such a way that your fingers would sweep into through the smaller angle between them.

4. Your thumb points in the direction of .

c

bac

a

b

c

a

b

27

Right Hand Rule for Vector Product:• By using the right-hand rule we can see that the

direction of , where

is opposite to the direction of . So,

that is:

as the figure shows.

Example: By using right-hand rule we find that the vector points in the direction, therefore:

c

abc

c

cc

abba

kkji ˆˆ90sin)1)(1(ˆˆ

ji ˆˆ k

28

Right Hand Rule for Vector Product:

29

• In unit vector notation we write:

Here each vector component of the first vector is to be crossed by each vector component of the second vector. Notice that:

and so on. By doing so we get:

This can be obtained using the determinant, as follows:

)ˆˆˆ()ˆˆˆ( kbjbibkajaiaba zyxzyx

kbakbajibaibia

iibaibia

yxyxyxyx

xxxx

ˆˆ)90)(sin1)(1()ˆˆ(ˆˆ

)00sinsin (Because 0)ˆˆ(ˆˆ

kabbajabbaiabbaba yxyxxzxzzyzyˆ)(ˆ)(ˆ)(

kabbajabbaiabba

bbb

aaa

kji

ba yxyxxzxzzyzy

zyx

zyxˆ)(ˆ)(ˆ)(

ˆˆˆ

30


31

3-7 Vectors and the Laws of Physics If we rotate the axes (but not the

vector ), through an angle as

in the figure the components will have

new values say,

Since there are infinite number of

choices of there are an infinite

number of different pairs of components of .

Each of these pairs produce the same magnitude and direction for the vector. In the figure, we have:

We thus have freedom in choosing the coordinate system. This is also true of the relations (laws) of physics; they are all independent of the choice of the coordinate system.

a

.' and' yx aa

a

' and '' 2222yyyx aaaaa

3-2 vectors and scalars is a number with units. it can be positive or negative. example: distance,...

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