Find the length of the line given by \( \mathbf{r}(\mathrm{t})=\langle 7 \mathrm{t}, 10 \mathrm{t}\rangle \) for \( \mathrm{a} \leq \mathrm{t} \leq \mathrm{b} \). The length of the line is (Type an expression using a and b as the variables. Type an exact answer, using radicals as needed.)

To find the length of the line given by \(\mathbf{r}(t) = \langle 7t, 10t \rangle\) for \(a \leq t \leq b\), we start by calculating the derivative of \(\mathbf{r}(t)\): \[ \mathbf{r}'(t) = \langle 7, 10 \rangle. \] The magnitude of the derivative \(|\mathbf{r}'(t)|\) gives the speed along the curve: \[ |\mathbf{r}'(t)| = \sqrt{7^2 + 10^2} = \sqrt{49 + 100} = \sqrt{149}. \] The length \(L\) of the line segment from \(t = a\) to \(t = b\) is given by the integral of the speed: \[ L = \int_a^b |\mathbf{r}'(t)| \, dt = \int_a^b \sqrt{149} \, dt. \] Since \(\sqrt{149}\) is a constant, we can simplify the integral: \[ L = \sqrt{149} \cdot (b - a). \] So, the length of the line is \[ \boxed{\sqrt{149} (b - a)}. \]