Multplication of inner products?

Question

I'm working on a inner product problem and I think that this inner product is true but I'm not sure:

$(m,n)(x,x)=(m,x)(n,x)$

where $m,n,x$ are all vectors. Thanks.

Answer 1

It's not. Let $m=n=(1,0)$ and $x=(0,1)$. Then $$(1,0)\cdot(1,0)=1\\(0,1)\cdot(0,1)=1\\(1,0)\cdot(0,1)=0$$ So $1\cdot1=0$. This is false.

Answer 2

Let $a,\,b$ be an angle $\theta_{ab}$ apart. For nonzero vectors, your claim is equivalent to $\cos\theta_{mn}=\cos\theta_{mx}\cos\theta_{nx}$. Or if you prefer an algebraic approach,$$m_in_ix_jx_j-m_ix_in_jx_j=m_ix_j(n_ix_j-x_in_j)=m_ix_j(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})n_kx_l.$$