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Home » \[ V = 3.14159 imes 3^2 imes 10 = 3.14159 imes 9 imes 10 \] - Crosslake

\[ V = 3.14159 imes 3^2 imes 10 = 3.14159 imes 9 imes 10 \] - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Using the identity $ \mathbf{a} \times (\mathbf{v} \times \mathbf{a}) = (\mathbf{a} \cdot \mathbf{a})\mathbf{v} - (\mathbf{a} \cdot \mathbf{v})\mathbf{a} $, and since $ \mathbf{a} \cdot \mathbf{v} = 0 $, this becomes:

📌 \|\mathbf{a}\|^2 \mathbf{v} = \mathbf{a} \times \mathbf{b}

📌 Compute $ \|\mathbf{a}\|^2 = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14 $

📌 Now compute $ \mathbf{a} \times \mathbf{b} $:

📌 \mathbf{a} \times \mathbf{b} = \begin{vmatrix}

📌 & -2 & 1 \\

📌 \end{vmatrix} = \mathbf{i}(2 \cdot 1 - 3 \cdot (-2)) - \mathbf{j}(1 \cdot 1 - 3 \cdot 1) + \mathbf{k}(1 \cdot (-2) - 2 \cdot 1)

📌 = \mathbf{i}(2 + 6) - \mathbf{j}(1 - 3) + \mathbf{k}(-2 - 2) = \langle 8, 2, -4 \rangle

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