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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 $\pi/3, 5\pi/3, 7\pi/3, 11\pi/3$, butEarn Big Fast: Unlock Your Walmart Affiliate Program and Start Making Money Today!Walmart Affiliate Program Secrets Revealed—Get Paid for Every Click!miR6. "How to Score Hidden Earnings with Walmart’s HOT Affiliate Program!"Join Walmart’s Affiliate Gang Now—Turn Clicks Into Cash Like a Pro!"Sleep on This: Walmart Affiliate Program Pays More Than You Think!wallet+Plus Cost Breakdown: Walmart Plus Saves You Thousands – Don’t Miss This!How Walmart Plus Splashing Its True Cost – You’ll Shock What You’ll Save!