Integral with dot product in it

[dingo_d]

Homework Statement

I need to evaluate this integral:

\int (\hat{r}\cdot\vec{a})\hat{r}d\Omega

where \hat{r} is the radial unit vector, \vec{a} is a constant vector and d\Omega is solid angle element (\sin\theta d\theta d\phi).

I saw something similar, but it was with tensors, and mean values but that was puzzling enough :\

I tried looking at components, but I came to nothing smart. I have the answer (\frac{4\pi}{3}\vec{a}).

Any hints?

 

[dingo_d]

Should I try to write everything in component form and directly integrate? I'll try that...

EDIT:

Ok I solved it! I needed to decompose everything in Cartesian coordinates and then I ended up with 9 double integrals, of which I had 3 that survived and they were all \frac{4\pi}{3} with different unit vectors, so that combined gave \vec{a}.

:)