Given the following lines, determine the values of \( a \) and \( b \) for which the lines are parallel. \( \begin{array}{l}\mathbf{r}_{1}=[-9,-3,-2]+t[a, 4,5] \\ \mathbf{r}_{2}=[0,2,-1]+t[14, b, 10] \\ a=\text { Ex: 43 } \\ b=\end{array} \)

For the lines \( \mathbf{r}_{1} \) and \( \mathbf{r}_{2} \) to be parallel, their direction vectors must be scalar multiples of each other. The direction vector of \( \mathbf{r}_{1} \) is \( [a, 4, 5] \), and the direction vector of \( \mathbf{r}_{2} \) is \( [14, b, 10] \). To express that one vector is a scalar multiple of the other, we can set up the following relationships: \[ \frac{a}{14} = \frac{4}{b} = \frac{5}{10} \] The last ratio simplifies to: \[ \frac{5}{10} = \frac{1}{2} \] From this, we can express \( 4 \) in terms of \( b \): \[ \frac{4}{b} = \frac{1}{2} \implies 4 = \frac{b}{2} \implies b = 8 \] Next, we use \( a \): \[ \frac{a}{14} = \frac{1}{2} \implies a = \frac{14}{2} = 7 \] Thus, the values of \( a \) and \( b \) for which the lines are parallel are: \[ a = 7,\ b = 8 \]