Define a pairing $\langle\cdot,\cdot\rangle$, based on the given matrix $A$, on $\textbf{R}^{3}\times\textbf{R}^{3}$ by $\langle v,w\rangle = v^{T}Aw$, where \begin{align*} A = \begin{bmatrix} 1 & 2 & 4\\ 3 & 0 & 5\\ 4 & 5 & 6 \end{bmatrix} \end{align*}
Does this just mean I have to pick two random numbers to complete the dot product?
This is defining an inner product of two vectors, chosen freely from $\Bbb R^3$, and the definition simply includes a fixed matrix $A$. You could think of it like a function of vectors in a sense:
$$f( \vec x,\vec y ) = \langle \vec x , \vec y \rangle = \vec x^T A \vec y$$
You don't "pick random numbers," but rather this definition applies for any vectors $\vec x, \vec y$ in $\Bbb R^3$. Moreover, you're not using a dot product; here, typical matrix multiplication is going on.