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📰 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 $\boxed{\dfrac{\pi}{6}}$. 📰 Question: A biochemistry technician measures the angle between two molecular bonds modeled as vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$. Compute $\cos\theta$ where $\theta$ is the angle between them. 📰 Solution: The cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$. Compute the dot product: $\mathbf{a} \cdot \mathbf{b} = (1)(0) + (0)(1) + (1)(1) = 1$. The magnitudes are $\|\mathbf{a}\| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}$ and $\|\mathbf{b}\| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{2}$. Thus, $\cos\theta = \frac{1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2}$. The final answer is $\boxed{\dfrac{1}{2}}$. 📰 Mason Alexander Indianapolis 4252422 📰 Astral Chain Revealed The Secret Power Behind Cosmic Energy Hacking 2172174 📰 Keilani Asmus 9104002 📰 This Flag Changed Puerto Ricoheres Why The Us Ignores It Over And Over 2648634 📰 Gamma Blue 11 Unleashed Secrets You Never Knew Exist 2235284 📰 Hotel Lime Light 9241026 📰 Crab Imperial Recipe 7986565 📰 Bioware Games 6154239 📰 Financial Services Solutions 5089125 📰 Unlock The Tbf Meaning Fastthis One Definition Will Change Everything 9348465 📰 This Unbelievable Breakthrough By Tom Lee Could Trigger Ethereums Supercycledont Miss 4545419 📰 You Wont Believe These Dino Hunting Secrets Youre Missing In 2024 9864211 📰 Hot Flashes Film 9715725 📰 Kentucky University Fees 977623 📰 Download The Ultimate Slime Game Nowunlimited Slime Battles Await 5107063