I have an expression $$p_m p_j (\delta_{ij} r_m + \delta_{jm} r_i + \delta_{im} r_j)$$
So if we expand things out $$=(\delta_{ij} r_m p_m p_j+ \delta_{jm} r_i p_m p_j+ \delta_{im} r_j p_m p_j)$$
Evaluating the $\delta$'s we are left with
$$=r_m p_m p_i + r_i p_j p_j + r_j p_i p_j$$
Which would be (I think)
$$=(\vec{r} \cdot \vec{p})\vec{p} + \vec{r}(\vec{p} \cdot \vec{p}) + \vec{p}(\vec{r} \cdot \vec{p})$$ Which is fine. However, I want to divide the original expression through by $p_j$, leaving
$$p_m (\delta_{ij} r_m + \delta_{jm} r_i + \delta_{im} r_j) $$ Again expanding $$=(\delta_{ij} r_m p_m + \delta_{jm} r_i p_m + \delta_{im} r_j p_m ) $$ $$=(\delta_{ij} r_m p_m + r_i p_j + r_j r_i)$$ Im not sure how to evaluate the delta in the final expression. Online sources I can find say $\delta_{ij}r_m p_m = r_i p_i$? Wouldn't this mean we're adding a tensor and scalar in the final expression? I'm confused.