Let us consider the spherical system with unit vectors $ r,\theta ,\phi $

Consider a vector $\vec { a } = { a }_{ r }\hat { r } +{ a }_{ \theta }\hat { \theta } +{ a }_{ \phi }\hat { \phi } $

$\vec { a } $ is a vector is space with some arbitrary direction.

Since vectors can be translated, $\vec { a } $ on translation to the origin will make all components other than those in $\hat{r}$ to be null. This would entirely change the original vector.

For a short example, a tangent to the circle lies along $ \hat {\theta}$ . On translating this to the origin, the vector becomes radial along $\hat{r}$

What is wrong here?

  • 1
    $\begingroup$ It would change the components because the basis vectors depend on position, but it wouldn't change the vector. $\endgroup$
    – Javier
    Aug 24 '17 at 18:43

The issue lies in the fact that $\hat{r}$, $\hat{\theta}$, and $\hat{\phi}$ are position dependent, so you cannot simply translate the vector $a$ and keep the components $a_{r}$, $a_{\theta}$ and $a_{\phi}$ constant. In fact, at the origin, these unit vectors are not defined.


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