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Math Help - a,b,c are vectors. How to prove vector b = c, given a.b = a.c and axb = axc?

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    a,b,c are vectors. How to prove vector b = c, given a.b = a.c and axb = axc?

    a,b,c are vectors. How to prove vector b = c, given a.b = a.c and axb = axc?
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    Re: a,b,c are vectors. How to prove vector b = c, given a.b = a.c and axb = axc?

    I think we need an additional assumption that \mathbf{a} \ne \mathbf{0}; otherwise the claim is not true.

    So assuming \mathbf{a} \ne \mathbf{0}, as a first step let's see if we can show that if \mathbf{a} \times \mathbf{v} = \mathbf{0} and \mathbf{a} \cdot \mathbf{v} = 0 then \mathbf{v} = \mathbf{0}.

    We have
    \mathbf{0} = \mathbf{v} \times (\mathbf{a} \times \mathbf{v}) = \mathbf{a} (\mathbf{v} \cdot \mathbf{v}) - \mathbf{v} (\mathbf{a} \cdot \mathbf{v}) =  \mathbf{a} (\mathbf{v} \cdot \mathbf{v}) - \mathbf{v} (0) =  \mathbf{a} (\mathbf{v} \cdot \mathbf{v})
    so
    \mathbf{v} \cdot \mathbf{v} = 0, hence \mathbf{v} = \mathbf{0}.

    Finally, to show that if \mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} and \mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} then \mathbf{b} = \mathbf{c}, let \mathbf{v} = \mathbf{a} - \mathbf{c} above.
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