Integrating certain functions over the unit sphere $\mathbb{S}^2$

Question

Let $ \mathbb{S}^2$ the unit sphere, and $ \vec a$, $ \vec b$ two constant vectors. I have to prove that: $$ \iint\limits_{\mathbb{S}^2} \langle \vec x , \vec a \rangle \langle \vec x , \vec b \rangle \, d \sigma= \frac43 π \langle \vec a , \vec b \rangle $$ and by using this to prove that: $$ \iint\limits_{\mathbb{S}^2} \langle A\vec x , \vec x \rangle \, d \sigma = \frac{4}{3} π \operatorname{tr}(A) $$ where $A$ is a matrix with order $3 \times 3$. Can anyone give me an idea about the solution?

Answer 1

$$ \iint\limits_{\mathbb{S}^2} x^2 + y^2 + z^2 d \sigma= 4 π, $$ but he integrals of $x^2,y^2,z^2$ must be the same, so $$ \iint\limits_{\mathbb{S}^2} x^2 d \sigma= 4 π / 3. $$ After that you are looking at the polarization identities for quadratic forms. Also called parallelogram law.