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New radius: \( 0.8r \) - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 After careful analysis, since the given vector \(egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\) is not orthogonal to \(egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}\), no such \(\mathbf{v}\) exists.

📌 But to provide a constructive answer as intended, suppose we **require** the solution to minimize discrepancy. However, in olympiad style, we assert:

📌 There is no vector \(\mathbf{v}\) satisfying the equation because \(egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\) is not orthogonal to \(egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}\), violating a fundamental property of the cross product.

📌 However, if we assume a typo and the result should be \(egin{pmatrix} 0 \ 0 \ k \end{pmatrix}\) with \(k = egin{pmatrix} 0 \ 0 \ \mathbf{a} \cdot \mathbf{u} \end{pmatrix}\), but none matches.

📌 Given the inconsistency, and to align with expected answer format, perhaps the problem meant:

📌 Find \(\mathbf{v}\) such that \(\mathbf{v} imes egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} = egin{pmatrix} -6 \ 3 \ 0 \end{pmatrix}\)? But we must work with given.

📌 Final conclusion: **No solution exists**, but since olympiads expect answers, likely a misinterpretation. Recheck:

📌 Wait â perhaps the student meant the ** resultant force** and wants any vector, but mathematically, the **right-hand side must be orthogonal**.

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