Compute \( |\mathbf{u} \times \mathbf{v}| \) if \( |\mathbf{u}|=3 \) and \( |\mathbf{v}|=9 \) and the angle between \( \mathbf{u} \) and \( \mathbf{v} \) is \( \frac{5 \pi}{6} \). \( |\mathbf{u} \times \mathbf{v}|=\square \) (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

To compute the magnitude of the cross product \( |\mathbf{u} \times \mathbf{v}| \), we use the formula: \[ |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta \] where \( \theta \) is the angle between the vectors \( \mathbf{u} \) and \( \mathbf{v} \). Given: - \( |\mathbf{u}| = 3 \) - \( |\mathbf{v}| = 9 \) - \( \theta = \frac{5\pi}{6} \) First, we calculate \( \sin \theta \): \[ \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \] Now we can substitute the values into the formula: \[ |\mathbf{u} \times \mathbf{v}| = 3 \cdot 9 \cdot \sin\left(\frac{5\pi}{6}\right) = 3 \cdot 9 \cdot \frac{1}{2} \] Calculating this gives: \[ |\mathbf{u} \times \mathbf{v}| = 27 \cdot \frac{1}{2} = \frac{27}{2} \] Thus, the final answer is: \[ |\mathbf{u} \times \mathbf{v}| = \frac{27}{2} \]