Find the angle \( \theta \), in degrees, between \( \mathbf{u} \) and \( \mathbf{v} \). \[ \begin{array}{l}\mathbf{u}=[-1,9] \\ \mathbf{v}=[5,-3] \\ \theta=\text { Ex: } 1.234^{\circ}\end{array} \]

To find the angle \( \theta \) between the vectors \( \mathbf{u} = [-1, 9] \) and \( \mathbf{v} = [5, -3] \), we can use the formula: \[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \] First, we calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \): \[ \mathbf{u} \cdot \mathbf{v} = (-1)(5) + (9)(-3) = -5 - 27 = -32 \] Next, we find the magnitudes of \( \mathbf{u} \) and \( \mathbf{v} \): \[ \|\mathbf{u}\| = \sqrt{(-1)^2 + 9^2} = \sqrt{1 + 81} = \sqrt{82} \] \[ \|\mathbf{v}\| = \sqrt{(5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \] Now, substituting these values into the cosine formula gives us: \[ \cos(\theta) = \frac{-32}{\sqrt{82} \cdot \sqrt{34}} \] Calculating the product \( \sqrt{82} \cdot \sqrt{34} = \sqrt{2788} \approx 52.785 \). Thus, \[ \cos(\theta) \approx \frac{-32}{52.785} \approx -0.606 \] To find \( \theta \), we take the inverse cosine: \[ \theta = \cos^{-1}(-0.606) \approx 127.0^\circ \] Therefore, the angle \( \theta \) between \( \mathbf{u} \) and \( \mathbf{v} \) is approximately: \[ \theta \approx 127.0^\circ \]