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تقرير عن vector analysis

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تقرير عن vector analysis

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

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Vectors




©️ 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 and
B
are 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 denoted
by -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, which
is obtained by placing the initial point of B on the final point
of A, and then drawing a line from the initial point of A
to the final point of B , as illustrated in Panel 4. This is sometines
referred to as the "Tip-to-Tail" method.


Panel 4

The operation of vector addition as described here can be written as
C
= 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 as
a 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 point
of -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 + AAssociative 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 similarly
      if 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: mA = Am
    Associative Law for Multiplication: (m + n)A = mA
    + nA, where m and n are two different scalars.
    Distributive Law: m(A + B) = mA + mB
    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 = Ax
+ Ay
. Where Ax and
Ay
are vectors in the x and y directions. If Ax and Ay
are the magnitudes of Ax and Ay, then
Ax and Ay
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. Notice
that the representation of A by it's components, Ax
and Ay
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 written
as F = -Fy,
but in the primed coordinate system F = -Fx'+
Fy'.
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, -Fx'which
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.
    Since A = Ax
    + Ay
    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.

    Dx = Ax + Bx + Cx

    Dy = Ay + By + Cy


    Panel 11
    Now you should use this simulation to study the very important topic
    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 angleq
    are given; what we wish to know is, what are Ax and Ay?


    Panel 12

    From elementary trigonometry we have, that cosq
    = Ax/|A| therefore Ax = |A| cos q,
    and similarly

    Ay = |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
    It is possible to define fundamental unit vectors in the Polar Coordinate
    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, A
and B denoted by A·B, is defined as the product of
the 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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