3-2 vectors and scalars is a number with units. it can be positive or negative. example: distance,...
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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
3-2 Vectors and Scalars
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
3-2 Vectors and Scalars
• 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
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��=��+ ��
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��tail
head
5
3-3 Adding Vectors Geometrically
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
3-3 Adding Vectors Geometrically
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.
3-3 Adding Vectors Geometrically
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− ��
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��− ����=��−��
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
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
3-8 Multiplying Vectors
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:
3-8 Multiplying Vectors
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
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
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
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