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election day 2025 results - Richter Guitar

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 \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}

📌 The magnitude of the perpendicular component (which is the desired quantity for speed component):

📌 \|\mathbf{v}_{\perp}\| = \sqrt{(2.5)^2 + 0^2 + (2.5)^2} = \sqrt{6.25 + 6.25} = \sqrt{12.5} = rac{5\sqrt{2}}{2}

📌 Alternatively, using formula for perpendicular component magnitude:

📌 \|\mathbf{v}_{\perp}\| = \sqrt{\|\mathbf{v}\|^2 - \left(rac{|\mathbf{v} \cdot \mathbf{c}|}{\|\mathbf{c}\|}

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