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Math Help - vectors

  1. #1
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    vectors

    I want to proof that, given vector a non zero and a vector b, so
    a.b=0 <=> \exists c: c\times a=b

    I know how to proof <=

    but how to proof =>?
    My try:
     a.b=0 => \exists c: c\times a=b
    <=> \sim \exists c: c\times a=b => \sim a.b=0
    <=> \forall c: c\times a \neq b => a.b\neq 0

    Supose \forall c: c\times a \neq b
    I tried to prove a.b \neq 0

    Now Supose \forall c: c\times a = b
    so b is orthogonal to both a and c
    so a.b=0

    So if \forall c: c\times a \neq b we have a.b \neq 0 my doubt is, my I affirm this???
    Last edited by Pipita; March 4th 2014 at 02:10 PM.
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  2. #2
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    Re: vectors

    Quote Originally Posted by Pipita View Post
    I want to proof that, given vector a non zero and a vector b, so
    a.b=0 <=> \exists c: c\times a=b
    I know how to proof <=
    but how to proof =>?
    Assume that $a\cdot b=0$ and $a\ne 0$. Now define $c=\dfrac{-(b\times a)}{||a||^2}$

    Does that work?
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  3. #3
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    Re: vectors

    I am at Lower class and learning about Mathematics so when many guyes replying here i will learn more thanks..
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  4. #4
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    Re: vectors

    Plato post #2 is correct. But define, out of the clear blue sky?

    axb is perpendicular to a and b and (axb)xa is then in direction b.

    (axb)xc=(a.c)b - (a.b)c from vector algebra, look it up.
    (axb)xa=(a.a)b - (a.b)c, which gives Plato's formula, c=axb/a2
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