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Math Help - properties of the zero vector

  1. #1
    Member billym's Avatar
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    properties of the zero vector

    Is a point "P" on a plane considered a zero vector, PP, or is there a difference between a single point and a zero vector?

    Can there be a polar representation of a single point? If so, would it be <0> ?
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  2. #2
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    Hello, billym!

    Interesting . . . "Zero" always raises strange questions.


    Is a point P on a plane considered a zero vector, \overrightarrow{PP}
    or is there a difference between a single point and a zero vector?
    There is always a difference between a point and a vector.

    The point might be (2,3) and the zero vector is: . \langle 0,0\rangle



    Can there be a polar representation of a single point?
    If so, would it be \langle 0\rangle ?
    If you referring to the pole (origin), the polar coordinates are: . (0,\theta)
    . . where the 'magnitude' is 0 (r=0) and \theta can be any angle.

    Other points have coordinates such as: . \left(2,\,\frac{\pi}{6}\right)

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  3. #3
    Member billym's Avatar
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    I'm still confused... what would be the polar representation of of a point with coordinates (-3,4)?
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  4. #4
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    Quote Originally Posted by billym View Post
    I'm still confused... what would be the polar representation of of a point with coordinates (-3,4)?
    x = r \cos \theta
    y = r \sin \theta

    If you want to switch form Cartesian to Polar

    note that x^2 + y^2 = r^2
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  5. #5
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    In terms of direction, do you treat the point as if it were a position vector, and then just state the magnitude as zero?
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  6. #6
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    It would be quite fair if someone argued that yours is a philosophical rather than a mathematical question.
    Vectors are hybrid objects in that a vector has both length and direction.
    Here is a quotation from Melvin Hausner, a geometer at NYU, “The vector \overrightarrow {PP} will be designated by 0 (read: zero, or the zero vector, depending on how fussy you are.) Thus \overrightarrow {PA}  = 0 is a way of writing P = A.”

    To answer your other question: \left( { - 3,4} \right) \equiv \left( {5,\pi  + \arctan \left( {\frac{{ - 4}}{3}} \right)} \right).
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  7. #7
    Member billym's Avatar
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    but I'm trying to find the polar representation of just a point (-3,4), how could a point have magnitude of 5?
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  8. #8
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    Quote Originally Posted by billym View Post
    but I'm trying to find the polar representation of just a point (-3,4), how could a point have magnitude of 5?
    Spend some time reading this, Polar coordinate system - Wikipedia, the free encyclopedia
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