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Understanding displacements of points by interpreting directions

  1. Nov 10, 2012 #1
    Suppose that points x and y are given in Euclidean space. Point x is displaced to point x1 by

    x1=x+a(x-y)

    Given that a is positive number, how can it be shown that the distance x1 to y is larger than distance x to y. I'm mainly interested in a vector interpretation of the above update rule. In that sense, can (x-y) in the above rule be interpreted as
    direction (force) from point y to point x? Similar interpretation is welcome.

    Given negative a, the update is x1=x+a(y-x), so now the above intuition of a force is direction from point x to point y.
     
  2. jcsd
  3. Nov 10, 2012 #2

    tiny-tim

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    hi onako! :smile:
    yes, that's exactly correct :smile:

    x-y is the vector ##\vec{YX}##, with magnitude |YX| and direction from Y to X
     
  4. Nov 12, 2012 #3
    So, adding a(x-y) to x, means that the distance x to y changes depending on a: positive a implies increased distance, and negative a implies decreased distance? It's a bit "non-rigid" to state that a vector is added to a point.
     
  5. Nov 12, 2012 #4

    tiny-tim

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    ah, no, it isn't added to a point, it's added to the vector ##\vec{OX}## …

    ##\vec{OX} + a\vec{XY} = \vec{OX_1}## :smile:

    draw the triangle, and you'll see why! :wink:
     
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