Find the angle between the vectors $\begin{pmatrix} 4 \\ 4 \\ 7 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 4 \\ 1 \end{pmatrix},$ in degrees.
If $\theta$ is the angle between the vectors, then
\[\cos \theta = \frac{\begin{pmatrix} 4 \\ 4 \\ 7 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 4 \\ 1 \end{pmatrix}}{\left\| \begin{pmatrix} 4 \\ 4 \\ 7 \end{pmatrix} \right\| \left\| \begin{pmatrix} 1 \\ 4 \\ 1 \end{pmatrix} \right\|} = \frac{(4)(1) + (4)(4) + (7)(1)}{9 \cdot \sqrt{18}} = \frac{27}{27 \sqrt{2}} = \frac{1}{\sqrt{2}}.\]Hence, $\theta = \boxed{45^\circ}.$