The angle between two vectors in spherical coordinates

The angle between two vectors in spherical coordinates is depicted in the diagram below:

We have

$cos\gamma=\frac{\boldsymbol{\mathit{r}}_i\cdot\boldsymbol{\mathit{r}}_j}{\vert\boldsymbol{\mathit{r}}_i\vert\vert\boldsymbol{\mathit{r}}_j\vert}=\hat{r}_i\hat{r}_j\;\;\;\;\;\;\;\;(33)$

where $\hat{r}_i$ and $\hat{r}_j$ are the unit vectors of $\boldsymbol{\mathit{r}}_i$ and $\boldsymbol{\mathit{r}}_j$ , respectively.

Substituting eq182b (the unit vector in spherical coordinates) in eq33, noting that $\hat{x}\cdot\hat{y}=\delta_{xy}$ and rearranging gives

$cos\gamma=sin\theta sin\theta^{'} cos(\phi-\phi^{'})+cos\theta cos\theta^{'}\;\;\;\;\;\;\;\;(34)$

The angle between two vectors in spherical coordinates

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