English for physics - Quản lí Giáo dục - Nguyễn Thị Kim Anh - Thư viện Bài giảng điện tử

GROUP EP150-4R

English for Physics

UNIT 1: VECTOR
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VECTOR
Work?

A study of motion involves
the introduction of various
quantities that are used to
describe the physical world.
Examples
of
such
quantities include distance,
displacement,
speed,
velocity,
acceleration,
force, mass, momentum,
energy, work, power, etc….

Velocity?

Mass?
Energy?


VECTOR

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 All of these quantities can be divided into two
categories: vectors and
scalars.
READING
 A vector quantity is fully described by its
magnitude and direction.
 In contrast, a scalar quantity is fully described by
its magnitude
VECTORS
SCALARS
 The emphasis of this lecture is to understand some
fundamental properties of vectors and to apply
these properties in order to understand motion
and forces that occur in two dimensions.


VECTOR
1

Basic concepts of vector

2

Basic vector operations

3

The dot product of two vectors

4

Cross product of two vectors

5

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The scalar triple product of three vectors


BASIC CONCEPT OF VECTOR

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Graphical of a vector

BASIC CONCEPT OF VECTOR
r
eAB

uuu
r
AB
= uuu
r
AB

uuu
r uuu
r r
AB = AB .eAB

A unit vector

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BASIC CONCEPT OF VECTOR

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BASIC CONCEPT OF VECTOR

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c) Parallel vectors
•Two vectors (which may have different magnitudes)
are said to be parallel if they are parallel to the same
line.
•If two vectors point toward opposite direction, they
are called anti-parallel vector.
•In summary, two vectors are parallel if one vector is a
scalar multiples of other.


BASIC CONCEPT OF VECTOR

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d) Equal vectors
•The length or magnitude of a vector is the distance
between the initial and terminal points of the vector.
The length of a vector AB is denoted by , so
•If two vectors a and b have the same length and
direction, they are said to be equal and denoted by

•The vector that has the same magnitude as but
points toward opposite direction is called the
opposite vector of and denoted by . Each vector has a
unique opposite vector.


BASIC CONCEPT OF VECTOR

Negation vector

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BASIC VECTOR OPERATIONS
a) Sum of two vectors (Vector addition)

ur
F

uu
r
F1
uu
r
F2

The sum of two vectors

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BASIC VECTOR OPERATIONS

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BASIC VECTOR OPERATIONS
a) Sum of two vectors (Vector addition)

The triangle rule

The parallelogram rule

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BASIC VECTOR OPERATIONS

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a) Sum of two vectors (Vector addition)

Commutation:

r r r r
a+b = b+a

r r r r r r
Associative: (a + b) + c = a + (b + c )
Adding with a zero vector:

r r r r r
a+0 = 0+a = a


BASIC VECTOR OPERATIONS

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BASIC VECTOR OPERATIONS
c) Scalar multiple of a vector

so

As
ti
cia

ve

ve

iati

tive

tive
c ia

ib u

soc
As

so
As

tr
Dis

r
r
r r r
r r

r r
k (a + b) = k .a + k .b (h + k ).a = h.a + k .a h.(k.a) = (h.k ).a

r r
1.ar= a r
(−1).a = −a


THE DOT PRODUCT OF TWO VECTORS

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Definition:
r
r
Given two non-zero
vectors
a andb r, rthe dot
r
r
product of two vectorsa andbr ,r denoted
by
,
is
a
a
.
b

r r
scalar defined by ther formula:
θ
r a.b = a . b .cos θ ,
where is the angleabetween
and
b
ur
For example, in physics, we know that if a
F
force acts on an object
and moves it a distance s,
ur
then the work A of the
F force is calculated by the
ur r ur r
ur r
formula:
A = F .s = F . s .cos( F , s )


THE DOT PRODUCT OF TWO VECTORS

Commutation:

Properties

Distributive:

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rr rr
a.b = b.a
r r r urr r r
a.(b + c) = a.b + a.c

Scalar Mutiplication:

r r
rr r r
(k .a).b = k .(a.b) = a.(k .b)


CROSS PRODUCT OF TWO VECTORS

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Definition:
r
r
Given two non-zero
vectors
a andb r, the
r
r
r cross

product of two vectorsa andb r , denoted
a × b , is the
r r r by
ˆ
vector defined by the formula:
a
×
b
=
a
.
b
.sin(
θ
,
n
) ,
r
r
where θ is the angle betweena andbr n;ˆ is
r a unit
vector perpendicular plane of vector
a and
b
The choice between
directions
r two (opposite)
r
that are perpendicular toa both b and is
determined by the right hand rude.

The right – hand rule


CROSS PRODUCT OF TWO VECTORS
Properties
1

2

3

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THE SCALAR TRIPLE PRODUCT OF TREE VECTORS

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Definition:
The scalar triple product (also called the mixed
product, box product or triple scalar product) is
defined as the dot product of one of the vectors with
r r r
the cross product of the other
a.(b × ctwo:
)

Three vectors defining a parallelepiped


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THE SCALAR TRIPLE PRODUCT OF TREE VECTORS

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Definition:
In Cartesian coordinate, we have:

r
i

r
j

r
k

ax a y az

r r r
r r r
a.(b ∧ c) = (ax i + a y j + az k ) bx by bz = bx by bz
cx c y cz cx c y cz

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THE SCALAR TRIPLE PRODUCT OF TREE VECTORS

Circular shift

r r r r r r
a.(b × c) = b.(c × a)
r r r
= c.(a × b)

Properties

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Negates

r r r
r r r
a.(b × c) = −a.(c × b)
r r r
r r r
= b.(a × c) = −c.(b × a)


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THE SCALAR TRIPLE PRODUCT OF TREE VECTORS

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Properties
The scalar triple product can also be
understood as the determinant of the 3×3 matrix
(thus also its inverse) having the three vectors either
as its rows or its columns (a matrix has the same
determinant as its transpose):
a1 a2 a3
r r r
r r r

ữ
a.(b ì c) = det  b1 b2 b3 ÷ = det(a, b, c)
c c c ÷
2
3 
 1
r r r r r r r r r r r r
a.(a × b) = a.(b × a ) = a.(b × b) = b.(a × a ) = 0


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