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knock knock on heaven's door song - ECD Germany

📅 May 9, 2026 👤 scraface

May 09, 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 \theta

📌 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:

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