Problem

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

Solution

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: \(\boxed{28}\)

From Solvely APP
Source: https://solvelyapp.com/problems/34713/

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