# تقرير عن vector analysis

## تقرير عن vector analysis

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www.utdallas.edu/~cantrell/ee4301/vector-analysis.pdf

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Department of Physics, University of Guelph

In this tutorial we will examine some of

the elementary ideas concerning vectors. The reason for this introduction

to vectors is that many concepts in science, for example, displacement,

velocity, force, acceleration, have a size or magnitude, but also they

have associated with them the idea of a direction. And it is obviously

more convenient to represent both quantities by just one symbol. That is

the

This is often simplified to just .

The line and arrow above the Q are there to indicate that the symbol represents

a vector. Another notation is boldface type as:

Note, that since a direction is implied, .

Even though their lengths are identical, their directions are exactly opposite,

in fact

The magnitude of a vector is denoted by absolute value signs around

the vector symbol: magnitude of

The operation of addition, subtraction and multiplication of ordinary

algebra can be extended to vectors with some new definitions and a few

new rules. There are two fundamental definitions.

The operation of vector addition as described here can be written as

This would be a good place to try this simulation on the graphical

addition of vectors.

Vector subtraction is defined in the following

way. The difference of two vectors,

that is,

or

a vector addition.

Any quantity which has a magnitude but

no direction associated with it is called a

speed, mass and temperature.

Many of the laws of ordinary algebra hold

also for vector algebra. These laws are:

Vectors can be related to the basic coordinate

systems which we use by the introduction of what we call "

The vector

are vectors in the x and y directions. If A

are the magnitudes of

A

are the vector components of

The breaking up of a vector

into it's component parts is known as

that the representation of

and A

is not unique. Depending on the orientation of the coordinate system with

respect to the vector in question, it is possible to have more than one

set of components.

It is perhaps easier to understand this by having a look at an example.

In the unprimed coordinate system, the vector

as

but in the primed coordinate system

F

Which representation to use will depend on the particular problem that

you are faced with.

For example, if you wish to determine the acceleration of the block

down the plane, then you will need the component of the force which acts

down the plane. That is, -F

would be equal to the mass times the acceleration.

The breaking up of a vector into it's components,

makes the determination of the length of the vector quite simple and straight

forward.

The resolution of a vector into it's components

can be used in the addition and subtraction of vectors.

Until now, we have discussed vectors in

terms of a Cartesian, that is, an x-y coordinate system. Any of the vectors

used in this frame of reference were directed along, or referred to, the

coordinate axes. However there is another coordinate system which is very

often encountered and that is the

The multiplication of two vectors, is not

uniquely defined, in the sense that there is a question as to whether the

product will be a vector or not. For this reason there are two types of

vector multiplication.

Let us do an example. Consider two vectors,

and . Now what is the angle between

these two vectors?

This concludes our survey of the elementary properties

of vectors, we have concentrated on fundamentals and have restricted ourselves

to the discussion of vectors in just two dimensions. Nevertheless, a sound

grasp of the ideas presented in this tutorial are absolutely essential

for further progress in vector analysis.

www.utdallas.edu/~cantrell/ee4301/vector-analysis.pdf

اضافة للموضوع

VectorsVectors

Department of Physics, University of Guelph

This Vector tutorial has been selected by PSIgate as a recommended teaching tool. Click the PSIgate logo to access their large inventory of Science Tutorials. |

In this tutorial we will examine some of

the elementary ideas concerning vectors. The reason for this introduction

to vectors is that many concepts in science, for example, displacement,

velocity, force, acceleration, have a size or magnitude, but also they

have associated with them the idea of a direction. And it is obviously

more convenient to represent both quantities by just one symbol. That is

the

**vector**. Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude. This is shown in Panel 1. . If we denote one end of the arrow by the origin O and the tip of the arrow by Q. Then the vector may be represented algebraically by OQ. | Panel 1 |

This is often simplified to just .

The line and arrow above the Q are there to indicate that the symbol represents

a vector. Another notation is boldface type as:

**Q**.Note, that since a direction is implied, .

Even though their lengths are identical, their directions are exactly opposite,

in fact

**OQ = -QO**.The magnitude of a vector is denoted by absolute value signs around

the vector symbol: magnitude of

**Q = |Q|**.The operation of addition, subtraction and multiplication of ordinary

algebra can be extended to vectors with some new definitions and a few

new rules. There are two fundamental definitions.

#1 Two vectors, A andBare equal if they have the same magnitude and direction, regardless of whether they have the same initial points, as shown in Panel 2. | Panel 2 |

#2 A vector having the same magnitude as A but in the opposite direction to A is denotedby -A , as shown in Panel 3. | Panel 3 |

We can now define vector addition.The sum of two vectors, A and B, is a vector C, whichis obtained by placing the initial point of B on the final pointof A, and then drawing a line from the initial point of Ato the final point of B , as illustrated in Panel 4. This is sometinesreferred to as the "Tip-to-Tail" method. | Panel 4 |

The operation of vector addition as described here can be written as

**C**

= A + B= A + B

This would be a good place to try this simulation on the graphical

addition of vectors.

*Use the "BACK" buttion to return to this point.*Vector subtraction is defined in the following

way. The difference of two vectors,

**A - B**, is a vector**C**that is,

**C = A - B**or

**C = A + (-B)**.Thus vector subtraction can be represented asa vector addition.

The graphical representation is shown in Panel 5. Inspection of the graphical representation shows that we place the initial point of the vector -B on the final point the vector A ,and then draw a line from the initial point of A to the final pointof -B to give the difference C. | Panel 5 |

Any quantity which has a magnitude but

no direction associated with it is called a

**"scalar"**. For example,speed, mass and temperature.

- The product of a scalar, m say, times a vector

**A**, is another

vector,

**B**, where

**B**has the same direction as

**A**but

the magnitude is changed, that is,

**|B|**= m

**|A|**.

Many of the laws of ordinary algebra hold

also for vector algebra. These laws are:

**Commutative Law for Addition: A + B = B + A**

**Associative Law for Addition: A + (B + C) = (A + B) + C**

The verification of the Associative law is shown in Panel 6. If we add A and B we get a vector E. And similarlyif B is added to C , we get F . Now D = E + C = A + F. Replacing E with ( A + B) and F with (B + C), we get (A +B) + C = A + (B + C) and we see that the law is verified.Stop now and make sure that you follow the above proof. | Panel 6 |

**Commutative Law for Multiplication:**m

**A = A**m

**Associative Law for Multiplication:**(m + n)

**A**= m

**A**

+ n

**A**, where m and n are two different scalars.

**Distributive Law:**m

**(A + B)**= m

**A**+ m

**B**

These laws allow the manipulation of vector quantities in much the same

way as ordinary algebraic equations.

Vectors can be related to the basic coordinate

systems which we use by the introduction of what we call "

**unit vectors**."- A unit vector is one which has a magnitude of 1 and is often indicated

by putting a hat (or circumflex) on top of the vector symbol, for example .The

quantity is

read as "a hat" or "a unit".

Let us consider the two-dimensional (or x, y)Cartesian Coordinate System, as shown in Panel 7. | Panel 7 |

We can define a unit vector in the x-direction by or it is sometimes denoted by . Similarly in the y-direction we use or sometimes . Any two-dimensional vector can now be represented by employing multiples of the unit vectors, and , as illustrated in Panel 8. | Panel 8 |

The vector

**A**can be represented algebraically by**A = A**

+ A. Where_{x}+ A

_{y}**A**and_{x}**A**_{y}are vectors in the x and y directions. If A

_{x}and A_{y}are the magnitudes of

**A**and_{x}**A**, then_{y}A

_{x}and A_{y}are the vector components of

**A**in the x and y directions respectively.The actual operation implied by this is shown in Panel 9. Remember (or ) and (or ) have a magnitude of 1 so they do not alter the length of the vector, they only give it its direction. | Panel 9 |

The breaking up of a vector

into it's component parts is known as

**resolving**a vector. Noticethat the representation of

**A**by it's components, A_{x}and A

_{y}is not unique. Depending on the orientation of the coordinate system with

respect to the vector in question, it is possible to have more than one

set of components.

It is perhaps easier to understand this by having a look at an example.

Consider an object of mass, M, placed on a smooth inclined plane, as shown in Panel 10. The gravitational force acting on the object is F = mg where g is the acceleration due to gravity. | Panel 10 |

In the unprimed coordinate system, the vector

**F**can be writtenas

**F**= -F_{y},but in the primed coordinate system

**F**= -F_{x'}+F

_{y'}.Which representation to use will depend on the particular problem that

you are faced with.

For example, if you wish to determine the acceleration of the block

down the plane, then you will need the component of the force which acts

down the plane. That is, -F

_{x'}whichwould be equal to the mass times the acceleration.

The breaking up of a vector into it's components,

makes the determination of the length of the vector quite simple and straight

forward.

- Since

**A**= A

_{x}

+ A

_{y}

then using Pythagorus' Theorem

.

For example

.

The resolution of a vector into it's components

can be used in the addition and subtraction of vectors.

- To illustrate this let us consider an example, what is the sum of the

following three vectors?

By resolving each of these three vectors into their components we see that the result is Panel 11. D _{x} = A_{x} + B_{x} + C_{x}D _{y} = A_{y} + B_{y} + C_{y} | Panel 11 |

of the algebraic

addition of vectors.

*Use the "BACK" buttion to return to this point.*

Very often in vector problems

you will know the length, that is, the magnitude of the vector and you

will also know the direction of the vector. From these you will need to

calculate the Cartesian components, that is, the x and y components.

The situation is illustrated in Panel 12. Let us assume that the magnitude of A and the angleqare given; what we wish to know is, what are A _{x} and A_{y}? | Panel 12 |

From elementary trigonometry we have, that cosq

= A

_{x}/

**|A|**therefore A

_{x}=

**|A|**cos q,

and similarly

A

_{y}=

**|A|**cos(90 - q) =

**|A|**

sinq.

Until now, we have discussed vectors in

terms of a Cartesian, that is, an x-y coordinate system. Any of the vectors

used in this frame of reference were directed along, or referred to, the

coordinate axes. However there is another coordinate system which is very

often encountered and that is the

**Polar Coordinate System.**In Polar coordinates one specifies the length of the line and it's orientation with respect to some fixed line. In Panel 13, the position of the dot is specified by it's distance from the origin, that is r, and the position of the line is at some angle q, from a fixed line as indicated. The quantities r and q are known as the Polar Coordinates of the point. | Panel 13 |

system in much the same way as for Cartesian coordinates. We require that

the unit vectors be perpendicular to one another, and that one unit vector

be in the direction of increasing r, and that the other is in the direction

of increasing q.

In Panel 14, we have drawn these two unit vectors with the symbols and . It is clear that there must be a relation between these unit vectors and those of the Cartesian system. | Panel 14 |

These relationships are given in Panel 15. | Panel 15 |

The multiplication of two vectors, is not

uniquely defined, in the sense that there is a question as to whether the

product will be a vector or not. For this reason there are two types of

vector multiplication.

- First, the

**scalar**or

**dot product**of two vectors, which

results in a scalar.

And secondly, the

**vector**or

**cross product**of two vectors,

which results in a vector.

In this tutorial we shall discuss only the scalar or dot product.

The scalar product of two vectors, Aand B denoted by A·B, is defined as the product ofthe magnitudes of the vectors times the cosine of the angle between them, as illustrated in Panel 16. | Panel 16 |

- Note that the result of a dot product is a scalar, not a vector.

The rules for scalar products are given in the following list, .

And in particular we have ,

since the angle between a vector and itself is 0 and the cosine of 0 is

1.

Alternatively, we have ,

since the angle between andis

90º and the cosine of 90º is 0.

In general then, if

**A·B**= 0 and neither the magnitude

of

**A**nor

**B**is 0, then

**A**and

**B**must be perpendicular.

The definition of the scalar product given earlier, required a knowledge

of the magnitude of

**A**and

**B**, as well as the angle between

the two vectors. If we are given the vectors in terms of a Cartesian representation,

that is, in terms of and ,

we can use the information to work out the scalar product, without having

to determine the angle between the vectors.

If, ,

then .

Because the other terms involved,,

as we saw earlier.

Let us do an example. Consider two vectors,

and . Now what is the angle between

these two vectors?

- From the definition of scalar products we have .

But .

This concludes our survey of the elementary properties

of vectors, we have concentrated on fundamentals and have restricted ourselves

to the discussion of vectors in just two dimensions. Nevertheless, a sound

grasp of the ideas presented in this tutorial are absolutely essential

for further progress in vector analysis.

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