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Math Help - vector length of magnitude

  1. #1
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    vector length of magnitude

    as far as I remember vector length of magnitude is same? however this sentence confuses me and I would appreciate if someone could explain it to me...

    "A vector can be represented by a section of a straight line, whose length is equal to the magnitude of the vector, and whose direction represents the direction of the vector."

    ... hose length is equal to the magnitude of the vector ... what does it mean?
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  2. #2
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    "Magnitude" of a vector is equal to its "length" if you are thinking about vectors geometrically- say, as "arrows". But there are many other kinds of vectors- for example, the set of all quadratic polynomials forms a vector space. We can define the "magnitude" of such vectors once we have a basis, but it wouldn't really make sense to call it a "length". If I took as basis \{1, x, x^2\}, then the magnitude of the vector ax^2+ bx+ c would be \sqrt{a^2+ b^2+ c^2}

    Here, then, they are talking about two different kinds of thing- vectors and line segments. One has a "magnitude", the other a "length". And all that sentence is saying is that you can do what you are doing automatically- representing a vector as a directed line segment.

    (I was taken aback for a moment by "hose length" but I think you just dropped the "w"!)
    Last edited by HallsofIvy; March 2nd 2011 at 07:23 AM.
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  3. #3
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    Sorry it's not clear! Maybe I am kind of a bad newbie ...
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  4. #4
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    In the first approximation, vector length and magnitude are the same.

    "A vector can be represented by a section of a straight line, whose length is equal to the magnitude of the vector, and whose direction represents the direction of the vector."

    ... whose length is equal to the magnitude of the vector ... what does it mean?
    This quote says that the magnitude is equal to the length, so it does not contradict what you know. If it said, "whose length is equal to half of the magnitude of the vector", that would raise a legitimate question.

    Going deeper, a vector is a more abstract object than a segment with designated beginning and end. For example, one can refer to pairs of real numbers (x,\alpha) where x\ge0, 0\le\alpha<2\pi as (two-dimensional) vectors. Here x is the magnitude of a vector and \alpha is the angle the vector forms with the horizontal axis. A pair of numbers is not literally a line segment, and the first number does not have to be interpreted as the length of the segment. However, there is a natural correspondence between such number pairs and line segments.

    Vectors can be even more abstract (e.g., polynomials), where it is not clear how to interpret them in Euclidean space. However, even in those cases vectors often can be associated with a real number called a magnitude.

    In any case, this is not a big deal. There is nothing wrong, when talking about vectors in Euclidean space, to think about them as line segments (up to a shift) and to identify their magnitudes and lengths.
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