Scattering geometry question While reading up on light scattering I came across this slide:

My vector maths is a bit rusty and I am having trouble understanding the last term (scattering geometry).
What is the significance of $\hat{r} \times \hat{E_{i}} \times \hat{r}$
 A: First of all the hats on $\hat r$ and $\hat E_i$ indicate that these are unit vectors.
$$(\vec A \times \vec B) \times \vec C \neq \vec A \times (\vec B \times \vec C)$$
See Vector Triple product 
Which means that $\vec A \times \vec B \times \vec C $ without parenthesis has no meaning. So I shall assume parenthesis around the first pair ie $(\vec A \times \vec B) \times \vec C $
$\vec A \times \vec B$ is given by the right hand rule and yields a vector $\vec Z$  perpendicular to both $\vec A$ and $\vec B$ and equal in magnitude to $\mid A\mid\cdot\mid B\mid \sin{\phi}$
If we cross this vector $\vec Z$ with $\vec A$ again we will get a vector perpendicular to $\vec A$ (obviously) but it also lies in the plane of $\{ \vec A,\vec B \}$. $\vec A$ and $\vec Z$ are perpendicular to each other so the $\sin{\phi}$ term is one and makes no contribution.
So to answer your question what is the significance of $\hat r \times \hat E_i \times \hat r$ gives you the direction that the field points and also adds a single $\sin{\phi}$ term to the magnitude.
A: you have three terms in the equation, which together define the scattered wave. the first two define the amplitude at any given time and place along the vector $\hat{r}$, which points the direction you're interested in.
$\hat{r}\times\hat{E}\times\hat{r}$ tells you that the wave is going to be perpendicular to $\hat{r}$ and also in the plane defined by two vectors $\hat{r},\hat{E}$
you picture is not good, shows only the angle $\theta$ to the direction of the light, it doesn't show the angle to $\hat{E}$. this angle is important, as if $\hat{r}$ is not perpendicular to $\hat{E}$, then the scattered wave will be smaller in amplitude. for instance, if $\hat{r}$ was vertical, i.e. parallel to $E$ in your picture, then the scattered wave would have zero amplitude.
