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Home » Double or Nothing 2024:CWhat Celebrities SKIPPED & Massive Twists You Can’t Miss! - Moon Smoking

Double or Nothing 2024:CWhat Celebrities SKIPPED & Massive Twists You Can’t Miss! - Moon Smoking

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Solution: The magnitude of the cross product is $\|\mathbf{p} \times \mathbf{q}\| = \|\mathbf{p}\| \|\mathbf{q}\| \sin\theta$. Since $\mathbf{p}$ and $\mathbf{q}$ are unit vectors, $\|\mathbf{p} \times \mathbf{q}\| = \sin\theta = \frac{1}{2}$. Thus, $\theta = \arcsin\left(\frac{1}{2}\right)$, which gives $\theta = \frac{\pi}{6}$ or $\frac{5\pi}{6}$. However, the angle between vectors is taken as the acute angle, so $\theta = \boxed{\dfrac{\pi}{6}}$.

📌 Question: A biochemistry technician studies protein folding angles, modeling bonds as vectors $\mathbf{m} = \begin{pmatrix} x \\ 2 \end{pmatrix}$ and $\mathbf{n} = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$. Find $x$ such that $\mathbf{m}$ and $\mathbf{n}$ are orthogonal.

📌 Solution: For vectors to be orthogonal, their dot product must be zero: $\mathbf{m} \cdot \mathbf{n} = x(1) + 2(-1) = x - 2 = 0$. Solving gives $x = 2$. The final answer is $\boxed{2}$.

📌 Question: A science policy analyst evaluates the spread of a technology using roots of $z^4 + z^2 + 1 = 0$. The maximum imaginary part of a root can be expressed as $\sin\theta$. Find $\theta$ in radians.

📌 Solution: Let $w = z^2$, so $w^2 + w + 1 = 0$. Solving gives $w = \frac{-1 \pm \sqrt{-3}}{2} = e^{\pm 2\pi i/3}$. Thus, $z^2 = e^{\pm 2\pi i/3}$, so $z = e^{\pm \pi i/3 + \pi i k/2}$ for $k = 0, 1$. The roots have angles $\pi/3, 5\pi/3, 7\pi/3, 11\pi/3$, but

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