1. **Problem statement:** Determine if vectors $\mathbf{A} = \begin{pmatrix}2 \\ 9 \\ 4\end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix}10 \\ 45 \\ 20\end{pmatrix}$ are collinear. 2. **Formula and rule:** Two vectors are collinear if one is a scalar multiple of the other, i.e., there exists a scalar $k$ such that $\mathbf{B} = k \mathbf{A}$. 3. **Check scalar multiples for each component:** $$k_1 = \frac{10}{2} = 5$$ $$k_2 = \frac{45}{9} = 5$$ $$k_3 = \frac{20}{4} = 5$$ 4. Since all three ratios are equal ($k_1 = k_2 = k_3 = 5$), the vectors are collinear. 5. **Conclusion:** Vector $\mathbf{B}$ is $5$ times vector $\mathbf{A}$, so $\mathbf{A}$ and $\mathbf{B}$ are collinear.