Answer
$3\sqrt{3}$
Work Step by Step
The dot product of two vectors $\mathbf a$ and $\mathbf b$ is $|\mathbf a||\mathbf b|\cos \theta $, where $\theta $ is the angle between the two vectors.
So if $|\mathbf a|=3$, $|\mathbf b|=\sqrt{6}$, and the angle between $\mathbf a$ and $\mathbf b$ is $45^\circ$, then
$$\mathbf a \cdot \mathbf b=|\mathbf a||\mathbf b|\cos (45^\circ)=3(\sqrt{6})(\frac{1}{\sqrt{2}})=3(\sqrt{3})=3\sqrt{3}.$$
