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Homework Help: Dot product

  1. Sep 13, 2006 #1
    Nice, simple, straighforward problem:
    [itex]j \cdot {-j}[/itex]

    Difficulty: I solved it two ways and got two answers.

    [itex]\vec{a} \cdot \vec{b}=ab\cos\phi[/itex]

    So in this case, I should get [tex]j \cdot {-j}=(1)(-1)(cos(180^o))=(1)(-1)(-1)=1[/itex]

    However, using the formula [itex]\vec{a} \cdot \vec{b}=a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z}[/itex], I got [itex]j \cdot {-j}=a_{y}b_{y}=(1)(-1)=-1[/itex]

    Can anyone tell me why?
     
    Last edited: Sep 13, 2006
  2. jcsd
  3. Sep 13, 2006 #2
    Recall that the dot product is:
    [tex] \vec A \cdot \vec B = |\vec A| |\vec B| \cos \theta_{AB} [/tex]

    let [itex] \vec A = (0,-1,0) [/itex]
    [tex] A=|\vec A| = \sqrt{(0)^2 +(-1)^2 +(0)^2} [/tex]

    What is [itex] (-1)^2 [/itex] :)
     
  4. Sep 13, 2006 #3
    Neither my book nor my professor mentions the absolute value thing. I guess that's completely implied, though, because the negative sign has to do with direction, not magnitude.

    Thank you so much!
     
  5. Sep 13, 2006 #4

    Astronuc

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    [tex] |\vec A|[/tex] means magnitude of [tex] \vec A[/tex], which is a scalar and is given by the square root of the sum of the squares of the magnitudes of the component vectors. [tex]|\vec{a}|\,=\,\sqrt{a^2_x+a^2_y+a^2_z}[/tex]



    One could also use the fact that -j * j = - (j * j) = - (1) = -1, where I use * to mean the dot product.

    Presumable j is a unit vector (0, 1, 0).

    Back to the OP:

    [tex]j \cdot {-j}=(1)(1)(cos(180^o))=(1)(1)(-1)=-1[/tex]
     
    Last edited: Sep 13, 2006
  6. Sep 13, 2006 #5
    I think basically your mistake lies in your calculation of the modulus(magnitude) of -j which is 1 not -1.For use in the formula a*b=abcos#
    where * means dot product
    a/b mean the magnitudes(moduli) of a & b
     
  7. Sep 13, 2006 #6
     
  8. Sep 13, 2006 #7
    The dot product operation has an associative property, that says:

    [tex] (k \vec A) \cdot \vec B = k (\vec A \cdot \vec B) [/tex]
    Where [itex] k [/itex] is a scalar, and [itex] \vec A, \,\, \vec B [/itex] are vectors.
     
    Last edited: Sep 13, 2006
  9. Sep 13, 2006 #8
    Riiiiiight....
    I've got to start consistently thinking of a negative sign as -1 multiplied by what follows, don't I.
    Thanks!
     
  10. Sep 14, 2006 #9
    I remember how much grief that used to give me :smile:
     
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