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Home » Total new volume: \( 20 + 10 = 30 \, \textliters \) - Crosslake

Total new volume: \( 20 + 10 = 30 \, \textliters \) - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 ext{proj}_{\mathbf{c}} \mathbf{v} = \left( rac{\mathbf{v} \cdot \mathbf{c}}{\mathbf{c} \cdot \mathbf{c}}

📌 ight) \mathbf{c}

📌 First, compute the dot products:

📌 \mathbf{v} \cdot \mathbf{c} = (2)(1) + (-1)(2) + (3)(-1) = 2 - 2 - 3 = -3

📌 \mathbf{c} \cdot \mathbf{c} = 1^2 + 2^2 + (-1)^2 = 1 + 4 + 1 = 6

📌 ext{proj}_{\mathbf{c}} \mathbf{v} = rac{-3}{6} egin{pmatrix} 1 \ 2 \ -1 \end{pmatrix} = -rac{1}{2} egin{pmatrix} 1 \ 2 \ -1 \end{pmatrix} = egin{pmatrix} -0.5 \ -1 \ 0.5 \end{pmatrix}

📌 Now compute the perpendicular component:

📌 \mathbf{v}_{\perp} = egin{pmatrix} 2 \ -1 \ 3 \end{pmatrix} - egin{pmatrix} -0.5 \ -1 \ 0.5 \end{pmatrix} = egin{pmatrix} 2.5 \ 0 \ 2.5 \end{pmatrix}

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