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 $$\vec{a}$$.

:)