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

Answer

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Answer

Final Answer: $\boxed{28}$

Steps

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: $\boxed{28}$