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Dot product and vectors

  1. Nov 14, 2007 #1
    1. The problem statement, all variables and given/known data
    I get confused with this problems
    show that the vector (orth of b onto a) = (b - proj of b onto a) is orthogonal to a.

    2. Relevant equations

    3. The attempt at a solution
    (b-proj of b onto a) dot a = 0
    and I got stuck!
  2. jcsd
  3. Nov 14, 2007 #2

    matt grime

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    Perhaps if you wrote out what "proj of b onto a" was in vector terms it would become easier.
  4. Nov 14, 2007 #3
    For example, I say:
    (b- a(a.b)/a^2).a=0
    How can I do next?
  5. Nov 14, 2007 #4


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    Staff Emeritus
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    You are not being careful to distinguish between vectors and numbers. The first "a" of "a(a.b)/a^2" is a vector while "a^2" is a number- the square of the length of a. You are trying to cancel them!

    with the result that you get this, which makes no sense! Does "(a.b)/a" mean you are dividing by a vector?

    [tex]\left(\vec{b}- \frac{\vec{a}\cdot\vec{b}}{|a|^2}\vec{a}\right)\cdot\vec{a}[/tex]
    [tex]\vec{b}\cdot\vec{a}- \frac{\vec{a}\cdot\vec{b}}{|a|^2}(\vec{a}\cdot\vec{a})[/tex]
    Now, what is that equal to?
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