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Home » Moldavite and Its Forbidden Magic: Is This Stone the Key to Otherworldly Transformation? - Crosslake

Moldavite and Its Forbidden Magic: Is This Stone the Key to Otherworldly Transformation? - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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To find $\cos \theta$, we use the formula for the dot product:
\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \thet...
First, compute the dot product $\mathbf{u} \cdot \mathbf{v}$:
\mathbf{u} \cdot \mathbf{v} = 3(-1) + 4(2) = -3 + 8 = 5
Next, find the magnitudes of $\mathbf{u}$ and $\mathbf{v}$:
\|\mathbf{u}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\|\mathbf{v}\| = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
Now, substitute these into the formula:
= 5 \cdot \sqrt{5} \cdot \cos \theta
Solve for $\cos \theta$:

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