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Vector definition

  1. Dec 12, 2015 #1
    Why we can not define a vector as a quantity which has magnitude and direction? Why we define the vectors according to behavior of its components in rotated coordinate-frames?
  2. jcsd
  3. Dec 12, 2015 #2


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    Vectors are, by definition, the elements of a vector space. In my opinion, other definitions are corollaries in disguise.
  4. Dec 12, 2015 #3


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    Vectors and tensors in general are mathematically defined independent of coordinate systems. This emphasizes the fact that they are geometrical objects which don't care about your preference for a certain coordinate system. If you regard a vector via its components as an array {a,b,...} which you can assign a length to, it looks like vectors cannot exist without the choice of coordinates.

    From the coordinate-free definition it follows that the components (!) have a certain behaviour under coordinate transformations, like the rotations you mentioned.
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