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: Properties of Vectors

  1. #1

    Jul 2011
    -
    15,678

    Properties of Vectors


    Vector addition
    ɡ .
    A B R
    (R= A + B---> (1.5
    : .
    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
    .
    : R=|A||B
    A&B

    Panel 2 #2 A vector having the same magnitude as A but in the opposite direction to A is denoted by -A , as
    .
    . .
    .
    R=A-B
    B= -A:.
    R=A-A=0<=


    2) : .
    .
    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
    A+B = C
    C ( ) .
    .
    .
    3) : .

    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.
    : .
    ..........
    .
    (A + B = B + A---> (1.6


  2. #2

    Jul 2011
    -
    15,678

    : Properties of Vectors

    Component of vector
    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.
    A x,y x y .
    A :



    Ax=A cosq

    Ay=A sinq
    :



    A, B, C, D , ........ (x,y) x y : x y




    :





    Sothat,
    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.


  3. #3

    Jul 2011
    -
    15,678

    : Properties of Vectors

    The unit vector
    .

    A A a

    A= a A (1.10)
    (i, j, k) rectangular coordinatesystemx, y, z :-




    ɡ A x,y



    : A B :






    Example
    Find the sum of two vectors A and B given by
    and

    Solution
    Note that Ax=3, Ay=4, Bx=2, and By=-5

    The magnitude of vector R is

    The direction of R with respect to x-axis is.


    Product of a vector
    ɡ .










    The scalar product
    scalar product dot product ɡ 0 90 90 180 90.



    .
    (1.16)
    :


  4. #4

    Jul 2011
    -
    4,543

    : Properties of Vectors

    Thanks Tamara
    SARAH

  5. #5

    : Properties of Vectors



  6. #6

    Jul 2012
    5,302

    : Properties of Vectors


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