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B What are the values in a vector?

  1. Feb 26, 2016 #1
    I'm trying to understand the concept of vectors. Vectors have magnitude and a direction. When I read vector with some values
    [tex]
    \textbf{x} = \left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right) = \left(\begin{array}{c}1\\2\\3\end{array}\right)
    [/tex]
    I'm not sure what these values are. Are the values points in the vector? Would I be wrong to imagine in a x,y graph, the values in this matrix are the values of y?
     
  2. jcsd
  3. Feb 26, 2016 #2

    jedishrfu

    Staff: Mentor

    This diagram may help:

    555px-3D_Vector.svg.png

    So the vector a has components a1, a2, and a3

    This is the most common usage. However, it doesn't have to be an orthogonal set of axes. It depends on the problem and the notational conventions you're using for describing your vector.

    By using a numbered set of axes you can generalize things to n-dimensional space.
     
  4. Feb 26, 2016 #3
    as your vectors are written as a column matrix with three elements it should be a three dimensional vector-then you can interpret the numbers!
     
  5. Feb 26, 2016 #4
    The diagram is super helpful!!!!!!
    This makes sense since x1 + x2 + x3 is 3 dimensional.
     
  6. Feb 26, 2016 #5

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The picture represents a 3 dimensional object. The arrow representing the vector is 3 dimensional. ##x_1+x_2+x_3## is just a number.
     
  7. Feb 28, 2016 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    A vector written as you have it make no sense without first having said what those number mean! A vector is defined by its 'direction and length'. A vector having a fixed direction and length 1 can be written as (1, 0, 0) or (0, 1, 0) or (0, 0, 1) or [itex]\left(\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right), etc.[/itex] depending upon what basis or coordinate system it is given in.
     
  8. Feb 29, 2016 #7

    Ssnow

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    Gold Member

    Vectors are element of a vector space ##V##. The axioms of a vector space are given after fixed the set of scalars (or generally a field) ##\mathbb{K}##. This can be ##\mathbb{R},\mathbb{C}, ...##. The components of your vectors are elements of ##\mathbb{K}## ...
     
  9. Mar 3, 2016 #8
    First take the 2 dimension case. Imagine the a baseball diamond where x1 = x , X2 = y.
    For any given pair of points (x,y) in the xy plane, a line segment can be drawn in the x direction of length x
    and another line segment can be drawn of length y in the y direction. With the tails of each line segment
    touching, a new line segment can be drawn between the x and y line segments starting at the adjoining tails
    Who's length is the sum of lengths of each x and y line segment. The angle (direction in the plane) of thIs
    vector can be found using x and y and trigonometry. Since this vector is bounded sorry far by by x,y only, now
    introduce x3 = z. Raise this vector the length of z units upward or downwards. This now defines the vector in 3 space
    that is bounded by the 3 planes xy, xz, and yz. The angle of the vector relative to xz or yz is the re-application
    of trigonometry using (x,z) or (y,z).
     
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