0
$\begingroup$

What can be the formula for dot product in spherical coordinates? that is $$\overrightarrow{A}(r, \theta , \phi). \overrightarrow{B}(r, \theta , \phi)=?$$ $$or,(A_r \hat{r} + A_{\theta} \hat{\theta} + A_{\phi} \hat{\phi}).(B_r \hat{r} + B_{\theta} \hat{\theta} + B_{\phi} \hat{\phi})=?$$

My Approach:
As $\overrightarrow{a}.(\overrightarrow{b} + \overrightarrow{c})=\overrightarrow{a}.\overrightarrow{b} + \overrightarrow{a}.\overrightarrow{c}$ $$\therefore (A_r \hat{r} + A_{\theta} \hat{\theta} + A_{\phi} \hat{\phi}).(B_r \hat{r} + B_{\theta} \hat{\theta} + B_{\phi} \hat{\phi})= A_r \hat{r}.(B_r \hat{r} + B_{\theta} \hat{\theta} + B_{\phi} \hat{\phi})+A_{\theta} \hat{\theta}.(B_r \hat{r} + B_{\theta} \hat{\theta} + B_{\phi} \hat{\phi})+A_{\phi} \hat{\phi}.(B_r \hat{r} + B_{\theta} \hat{\theta} + B_{\phi} \hat{\phi})$$ $$=A_r B_r + A_{\theta}B_{\theta} + A_{\phi}B_{\phi} \dots \dots (1)$$ $$\text{as } \hat{r}.\hat{r}=1; \hat{r}.\hat{\theta}=0;\hat{r}.\hat{\phi}=0 \space \& \space \hat{\theta}.\hat{\phi}=0 \text{ and so on...}$$ but i am not sure whether I am correct or wrong
so please help me...

$\endgroup$
1
$\begingroup$

You're right. The basis $\{\hat{e}_i\}_i= \{\hat{r},\hat{\theta},\hat{\phi} \}$ is orthonormal, so it follows

$$ \hat{e}_i\cdot \hat{e}_j = \delta_{ij} $$

$\endgroup$
  • $\begingroup$ do you mean $\{\hat{r},\hat{\theta},\hat{\phi} \}$ are fixed (same for A and for B) ?? $\endgroup$ – G Cab Mar 10 '18 at 11:55
  • $\begingroup$ @GCab How is that implied by what I wrote? $\endgroup$ – caverac Mar 10 '18 at 11:57
  • $\begingroup$ I mean, what is $(1,0,0)\cdot(1,\pi/2,0)$ ? $\endgroup$ – G Cab Mar 10 '18 at 12:02
  • $\begingroup$ @GCab I'm sorry but I fail to understand what that has anything to do with the reply. Are un interested in knowing the inner product of the vectors of the spherical basis and a cartesian rectangular basis? $\endgroup$ – caverac Mar 10 '18 at 12:08
  • $\begingroup$ sorry for not putting it clear before, in fact I am wondering about the relation between the dot product you show in spherical wrt the corresponding product in cartesian coordinates $\endgroup$ – G Cab Mar 10 '18 at 13:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.