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

1. 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?

2. 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.