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Home » The final answer is \(\boxed-2\). - Blask

The final answer is \(\boxed-2\). - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Substitute \( p = 2 \) into (4):

📌 3(2) + q = 8 \implies 6 + q = 8 \implies q = 2

📌 Substitute \( p = 2 \), \( q = 2 \) into (1):

📌 + 2 + r = 10 \implies r = 6

📌 Thus, \( p = 2 \), \( q = 2 \), \( r = 6 \).

📌 \boxed{p = 2, \, q = 2, \, r = 6}

📌 \]Question: A tech entrepreneur in Athens designs a drone navigation system where the drone's path is represented by unit vectors $\mathbf{u}$ and $\mathbf{v}$ with an angle $\theta$ between them. If $\mathbf{u} \cdot \mathbf{v} = \frac{\sqrt{3}}{2}$, find $\theta$.

📌 Solution: The dot product of two unit vectors is $\mathbf{u} \cdot \mathbf{v} = \cos\theta$. Given $\cos\theta = \frac{\sqrt{3}}{2}$, the angle $\theta$ satisfies $\theta = \arccos\left(\frac{\sqrt{3}}{2}\right)$. This corresponds to $\theta = 30^\circ$ or $\frac{\pi}{6}$ radians. However, since cosine is positive in both the first and fourth quadrants, but angles between vectors are typically taken in $[0, \pi]$, the solution is $\boxed{\dfrac{\pi}{6}}$.

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