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

**TaTzRE66** · August 30th 2012, 11:03 PM

**awkward** · September 1st 2012, 07:14 AM

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

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