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Homework Help: Show the commutative property with dot product

  1. Sep 9, 2005 #1
    Hello everyone, does anyone know the proof of the dot products communative property (a)(b) = (b)(a) or any websites that show the dot products communative property? or other properties? Thanks! The book only shows the distributed property.
     
  2. jcsd
  3. Sep 9, 2005 #2

    quasar987

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    Science Advisor
    Homework Helper
    Gold Member

    Write a.b and then b.a in terms of the components or as |a||b|cosO. Commutativity of the dot products follows from commutativity of addition (resp. multiplication) in [itex]\mathbb{R}[/itex].
     
  4. Sep 9, 2005 #3

    Astronuc

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    Staff Emeritus
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    (a1 i + b1 j) [itex]\bullet[/itex] (a2 i + b2 j) =

    a1 i [itex]\bullet[/itex] a2 i + a1 i [itex]\bullet[/itex] b2 j + b1 j [itex]\bullet[/itex] a2 i + b1 j [itex]\bullet[/itex] b2 j =

    remember i [itex]\bullet[/itex] i = j [itex]\bullet[/itex] j = 1, and i [itex]\bullet[/itex] j = j [itex]\bullet[/itex] i = 0,

    then regroup and show something similar to

    a * b = b * a

    and this can be extended to 3 dimensions
     
  5. Sep 9, 2005 #4
    well, the dot product is a definition.

    as such

    [tex]
    \vec{a}\bullet\vec{b}=

    \left(
    \begin{array}{cc}
    a_x \\
    a_y\\
    a_z
    \end{array}
    \right)
    \bullet
    \left(
    \begin{array}{cc}
    b_x \\
    b_y\\
    b_z
    \end{array}
    \right)
    =a_xb_x + a_yb_x + a_zb_z
    [/tex]

    so, what is [tex]\vec{b}\bullet\vec{a}[/tex]?
     
  6. Sep 10, 2005 #5
    Cool so all I really need to show is this? or doesn't this prove it yet? Thanks for the replies everyone
    [tex]
    \vec{b}\bullet\vec{a}=

    \left(
    \begin{array}{cc}
    b_x \\
    b_y\\
    b_z
    \end{array}
    \right)
    \left(
    \begin{array}{cc}
    a_x \\
    a_y\\
    a_z
    \end{array}
    \right)
    \bullet

    =b_xa_x + b_xa_y + b_za_z
    [/tex]
     
  7. Sep 10, 2005 #6

    Doc Al

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    Staff: Mentor

    You need to show that
    [tex]\vec{b}\bullet\vec{a} = \vec{a}\bullet\vec{b}[/tex]

    Evaluate each side and compare. Make use of the commutivity of ordinary addition, as quasar987 advised.
     
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