How to write this vector? Say I have the following:
$$A_x = x~p_x,~~ A_y = y~p_y,~~ A_z = z~p_z$$
and I can write $\vec{r} = \langle x,y,z\rangle$ is a position vector and $\vec{p} = \langle p_x,p_y,p_z\rangle$ is a momentum vector. Then is it possible for me to write:
$\vec{A} = \vec{r}\vec{p}$? (There is no dot product here). If not, how can I write $\vec{A}$ as a function of $\vec{r}$ and $\vec{p}$?
 A: What you have is a dyadic or dyadic tensor is a second order tensor, have a look at https://en.wikipedia.org/wiki/Dyadics
Two familiar ways of multiplying vectors are he dot product (that returns a scalar) and the cross product (that returns a [pseudo]vector).
The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context: For two vectors ${x_i}, {y_j}$ the corresponding dyadic has elements $A_{i,j} = x_i y_j$. Note that $A_{i,j} \ne A_{j,i}$ and the elements you have are the diagonal components.
A: You can denote it in whatever way you want, as long as you know how to work with your notation. However the canonical way of writing this would be the following (in tensor form).
\begin{equation}
A_i=\delta_{ij}r_ip_j
\end{equation}
You can think of some sort of a matrix form for writing this, but this is the most basic one. 
For example, you can easily differentiate by some $k$ variable:
\begin{equation}
\partial_kA_i=\delta_{ij}p_j\partial_kr_i+\delta_{ij}r_i\partial_kp_j
\end{equation}
