Spherical Symmetry

Let ${\bf A}$ and ${\bf B}$ be constant Vectors. Define

$$Q\equiv 3({\bf A}\cdot\hat{\bf r})({\bf B}\cdot\hat{\bf r})-{\bf A}\cdot{\bf B}.$$

Then the average of $Q$ over a spherically symmetric surface or volume is

$$\left\langle{Q}\right\rangle{}=\left\langle{3\cos ^2\theta -1}\right\rangle{}({\bf A}\cdot {\bf B})=0,$$

since $\left\langle{3\cos^2\theta-1}\right\rangle{}=0$ over the sphere.