How I could prove that this vector is a unit vector of a basis?

Question

I need prove that $$ ê_j(P)=\frac{\frac{\partial \vec{r}}{\partial q_i}}{| \frac{\partial \vec{r}}{\partial q_i}|} $$ is a unit vector of another basis, I think I have to apply that the partial of $\vec{r}$ respect of $q_i$ give a vector with the direction of $q_i$ and it is unitary because we divide by its length but I dont know how do with math lenguage

Answer 1

Since $\| \alpha v \|$ = $\vert \alpha \vert \cdot \| v \|$ (it's the property from definition of norm or length) then $\| \hat{e}_j(P) \|= \left \| \frac{\frac{\partial \vec{r}}{\partial q_i}}{ \left \| \frac{\partial \vec{r}}{\partial q_i} \right \|} \right \| = \frac{\left \| \frac{\partial \vec{r}}{\partial q_i} \right \|}{\left \| \frac{\partial \vec{r}}{\partial q_i} \right \|} = 1$.