Let $\vec{a} ,\vec{b},\vec{c}$ linear Independent vectors that form and ordered basis B. And consider this basis non orthogonal and $\vec{r} = x \vec{a} +y\vec{b}+z\vec{c}$. This vector space is euclidian and Real. Prove that:
$\displaystyle x = \frac{\vec{r} \cdot \vec{b}\times\vec{c}}{\vec{a} \cdot \vec{b}\times\vec{c}}$
$$R=xA+yB+zC$$ Or,$$R.(B×C)=xA.(B×C)+yB.(B×C)+zC.(B×C)$$ Or,$$R.(B×C)=xA.(B×C)$$as other dot products will be zero.
Or,$$x=R.(B×C)/(A.(B×C))$$