Tài liệu Chapter 3: Vectors docx

In particular we will learn the following: Geometric vector addition and subtraction Resolving a vector into its components The notion of a unit vector Add and subtract vectors by co

Chapter 3 Vectors

In Physics we have parameters that can be completely described by a number and are known as “scalars” Temperature, and mass are such parameters Other physical parameters require additional information about direction and are known as “vectors” Examples of vectors are displacement, velocity and acceleration In this chapter we learn the basic mathematical language to describe vectors In particular we will learn the following:

Geometric vector addition and subtraction

Resolving a vector into its components

The notion of a unit vector

Add and subtract vectors by components

Multiplication of a vector by a scalar

The scalar (dot) product of two vectors The vector (cross) product of two vectors (3-1)

An example of a vector is the displacement vector which describes the change in position of an object as it

moves from point A to point B This is represented by an arrow that points from point A to point B The length of the arrow is proportional to the displacement magnitude The direction of the arrow indicated the displacement direction

The three arrows from A to B, from A' to B', and from A''

to B'', have the same magnitude and direction A vector can be shifted without changing its value if its length and direction are not changed

In books vectors are written in two ways:

Method 1: (using an arrow above) Method 2: a (using bold face print)

The magnitude of the vector is indicated by italic print: a

a 

(3-2)

Geometric vector Addition

Sketch vector using an appropriate scale Sketch vector using the same scale

Place the tail of at the tip of The vector starts from the tail of and terminates at the tip of

a b

s a b  









commutative

Negative of a given

Vector addition is

has the same magnitude

vecto

as but opposite direction

r

b

a b b a





(3-3)

Geometric vector Subtraction

 

 

We write:

From vector Then we add to vector

We thus reduce vector subtraction to vector addition which we know how to do

we find

d a b a b

b

b

a

d a 







Note: We can add and subtract vectors using the method of components

For many applications this is a more convenient method

(3-4)

A B

C

A component of a vector along an axis is the projection

of the vector on this axis For example is the projection of along the x-axis The component

is defined by drawing straight lines fr

x

x

a

om the tail and tip of the vector which are perpendicular to the x-axis

From triangle ABC the x- and y-components

of vector are given by the

cos ,

equations:

If we know

in s

a

a a

a





and we can determine and From triangle ABC we h

, tan

ave:

y x

x

y

x

y

a

a a

a



(3-5)

Unit Vectors

A unit vector is defined as vector that has magnitude equal to 1 and points in a particular direction

A unit vector is defined as vector that has magnitude equal to 1 and points

in a particular direction Unit vector lack units and their sole purpose is

to point in a particular direction The unit vectors along the x, y, and z axes are labeled , , and , respect i j ˆ ˆ k ˆ ive l y.

Unit vectors are used to express other vectors

For example vector can be written as:

The quantities and are called

ˆ ˆ

vect or component s tor

x

y

a a i

a

a i a j

a

a j

 







(3-6)

x O

y

a 

r 

b  Adding Vectors by Components

We are given two vectors and

We want to calculate the vector sum

The components and are given by the equations:

and

x y

a a i a j b b i b j

r r i r j











(3-7)

x O

y

a 

d 

b  Subtracting Vectors by Components

We are given two vectors and

We want to calculate the vector difference

The components and of are given by the equations:

and

x y

a a i a j b b i b j

d a b d i d j

d

d  







d  a  b

(3-8)

Multiplication of a vector by a scalar r esults in a new vector

The magnitude of the new vector is given by

Multiplying a Vecto

:

If 0 vector has the

r by a Scalar

| |b s

a

b











The S

same

calar

direction

Product o

as vector

If 0

f two vector has a direction opposite to that of vector

The scalar product of two vectors and is given by:

Vectors =

a

a b

b

















The scalar product of two vectors is also known as the product The scalar product in terms

of vector components is given by the equation:

cos

"dot"

= x x y y z z

ab





(3-9)

The vector product of the vectors and

is a vector The magnitude of is given

The Vector Pro

by the equation:

The direction of is perpend

duct of two Vectors

s

icular

in

c

c

c

c









right hand

to the plane P defined

by the vectors and The sense of the vector is given by the :

a Place the vectors and tail to tail

b Rotate in the plane P along

rule

the shortest an

c

a













gle

so that it coincides with

c Rotate the fingers of the right hand in the same direction

d The thumb of the right hand gives the sense of The vector product of two vectors is also known as

b

c





"cross

the " p roduct (3-10)

The vector components of vector are given by the equations:

,

The Vector Product in terms of Vector Components

,

a a i a j a k b b i b j b k c c i c j c k

c

 







Note: Those familiar with the use of determinants can use the expres

Note: The order of the two vectors in the cross product is importa

sion

n

t

x y z x

y z

i j k

a b a a a

b b b

b

 











a a b

(3-11)