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: \(\boxed{28}\)