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Tensor in D-dimensional space crosswise with 2 vectors

  1. Nov 24, 2015 #1
    I have 2 vectors ##\vec {V}=(v_1,v_2,v_3,v_4...v_D)## and ##\vec {X}=(x_1,x_2,x_3,x_4...x_D)## in D-dimensional euclidean space.
    I want a tensor ,which is crosswise with both of them.

    I think that the tensor is parallel with (D-2)-dimensional area, am I right?
    I do not know a lot about tensors ,so in answer please also explain me the notation that you used.
     
  2. jcsd
  3. Nov 24, 2015 #2

    fzero

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    It's not exactly clear what you want. Do you mean a rank 2 tensor, which we could think of as a matrix? In that case, we can have two possible conditions:

    1. ##\sum_j M_{ij} v_j =0##, or
    2. ##\sum_j v_j M_{ji}=0##.

    You might also want both of these to be satisfied. These conditions are independent unless ##M## is symmetric, ##M_{ij}=M_{ji}##.

    Either way, an analog of the Gram-Schmidt process allows you to start with any tensor ##M## and construct a new tensor ##M'## that satisfies one or both of these conditions. For instance, as long as ##|V|\neq 0##,
    $$ M'_{ij} = M_{ij} - \sum_k M_{ik} v_k \frac{v_j}{|V|^2}$$
    will satisfy ##\sum_j M'_{ij} v_j = 0##. The other combinations should be easy to work out.

    Hopefully this component notation is familiar, or at least obvious to you.
     
  4. Nov 25, 2015 #3
    I can´t explain ,what I mean ,in terms of tensor ,but I´ll try to explain it in terms of what I want it for:
    There are 4 vectors ##\vec V_1## ,##\vec V_2## ,##\vec X## and ##\vec F##.
    ##F⊥V_2##
    F is parallel with plane ,which is also parallel with vectors ##\vec X## and ##\vec V_1##

    I know only value of vectors ##\vec X## and ##\vec V_1##. My friend knows only value of vector ##\vec V_2##. I want a tensor that I could give to my friend so he could calculate value of ##\vec F## without knowing vectors ##\vec X## and ##\vec V_1## separately.
     
  5. Nov 26, 2015 #4
    Nb: by value I meant the vector itself not length of vector.

    I there any such tensor at all ,I wanted? If there is not, it would be useful to learn that there is not.
     
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