Find $\mathbf{u} \cdot \mathbf{v}$ where $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$. Round your answer to four decimals.
\[
\|\mathbf{u}\|=4,\|\mathbf{v}\|=12, \theta=\frac{5 \pi}{6}
\]

Step 1 ：Given that the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\) are 4 and 12 respectively, and the angle between them is \(\frac{5 \pi}{6}\).

Step 2 ：The dot product of two vectors is given by the formula \(\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\|\|\mathbf{v}\|\cos(\theta)\), where \(\theta\) is the angle between the vectors.

Step 3 ：Substitute the given values into the formula: \(\mathbf{u} \cdot \mathbf{v} = 4 \cdot 12 \cdot \cos(\frac{5 \pi}{6})\).

Step 4 ：Calculate the dot product: \(\mathbf{u} \cdot \mathbf{v} = -41.569219381653056\).

Step 5 ：Round the result to four decimal places: \(\mathbf{u} \cdot \mathbf{v} = -41.5692\).

Step 6 ：So, the dot product of the two vectors is \(\boxed{-41.5692}\).