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Confusion regarding dot product of vectors(row matrices)

  1. Oct 26, 2009 #1
    I’ve got a confusion. We know a 1x3 row matrix is a 3-vector i.e.
    x= [ a b c]
    Matrix x can be written in vector notation like x= a i + b j + c k
    where i, j, k are unit vectors along x,y & z axes.
    For dot product of
    x.x = a2 + b2 + c2 when x= a i + b j + c k

    But according to the matrix multiplication rule, multiplication of two matrices is possible only when column of 1st matrix = row of the 2nd matrix.
    So x.x = [ a b c] [ a b c] is not possible
    My questions are :
    (1) Both x= [ a b c] and x= a i + b j + c k are same vector.
    Then why this discrepancy happens?
    (2) Does really x.x exist when x = [ a b c]? Can we approach in any other way to define x.x when x = [ a b c] ?
    I’m novice at linear algebra. So it would be helpful for me if you can explain elaborately. I’m really at a loss about that confusion.
     
  2. jcsd
  3. Oct 26, 2009 #2

    tiny-tim

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    Hi mdnazmulh! :smile:

    Technically, the product for "matrix" vectors is an inner product, and one of the vectors must be a transpose vector (written as a column vector instead of a row vector, or abbreviated xT).

    So the inner product is xxT.

    For details, you could see http://en.wikipedia.org/wiki/Inner_product_space" [Broken] …

    but I wouldn't bother until your professor deals with it. :smile:
     
    Last edited by a moderator: May 4, 2017
  4. Oct 26, 2009 #3
    Thanks for your reply. Actually in the book of introductory linear algebra by Bernard Kolman there is a question that if x is an n-vector then is it possible that x.x can have negative value? And part (b) of the question says that if x.x=0 , what is x=0?
    Now I understand the author placed those questions in the exercise just to check our conception. Answer MUST BE that x.x is no way possible.
    Thank u again
     
  5. Oct 26, 2009 #4

    tiny-tim

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    No, that's not what I meant.

    For a vector that's just a vector, x.x is correct.

    Only if a vector is considered as a matrix (which I called '"matrix" vectors'), does the product have to be xxT

    that's what you originally asked about. :wink:
     
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