Find \( \mathbf{u} \cdot \mathbf{v} \). \[ \|\mathbf{u}\|=25,\|\mathbf{v}\|=15 \text {, and the angle between } \mathbf{u} \text { and } \mathbf{v} \text { is } 5 \pi / 6 \text {. } \]

To find the dot product \( \mathbf{u} \cdot \mathbf{v} \), we can use the formula: \[ \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos(\theta) \] where: - \( \|\mathbf{u}\| = 25 \) is the magnitude of vector \( \mathbf{u} \), - \( \|\mathbf{v}\| = 15 \) is the magnitude of vector \( \mathbf{v} \), - \( \theta = \frac{5\pi}{6} \) is the angle between the vectors. Firstly, calculate \( \cos\left(\frac{5\pi}{6}\right) \): \[ \cos\left(\frac{5\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \] Now, substitute the values into the dot product formula: \[ \mathbf{u} \cdot \mathbf{v} = 25 \times 15 \times \left(-\frac{\sqrt{3}}{2}\right) = 375 \times \left(-\frac{\sqrt{3}}{2}\right) = -\frac{375\sqrt{3}}{2} \] **Final Answer:** \[ \mathbf{u} \cdot \mathbf{v} = -\frac{375\,\sqrt{3}}{2} \]