Find $\mathbf{u} \cdot \mathbf{v}$, where $\theta$ is the angle between the vectors $\mathbf{u}$ and $\mathbf{v}$. \[ \|\mathbf{u}\|=7,\|\mathbf{v}\|=8, \theta=\frac{\pi}{3} \]

Step 1 ：We are given the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\) as 7 and 8 respectively, and the angle between them as \(\frac{\pi}{3}\).

Step 2 ：We can calculate the dot product of the vectors using the formula \(\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\|\|\mathbf{v}\|\cos(\theta)\).

Step 3 ：Substituting the given values into the formula, we get \(\mathbf{u} \cdot \mathbf{v} = 7 \times 8 \times \cos(\frac{\pi}{3})\).

Step 4 ：Solving this gives us \(\mathbf{u} \cdot \mathbf{v} = 28\).

Step 5 ：Final Answer: \(\boxed{28}\)