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Competitors at the 2001 Summer Universiade - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 A counterclockwise rotation by \(90^\circ\) is represented by:

📌 egin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}

📌 Scaling by 2 multiplies the entire transformation matrix by 2:

📌 \mathbf{M} = 2 egin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} = egin{pmatrix} 0 & -2 \ 2 & 0 \end{pmatrix}

📌 Thus, the required matrix is \(oxed{egin{pmatrix} 0 & -2 \ 2 & 0 \end{pmatrix}}\).

📌 A civil engineer analyzing directional stability in solar panel arrays needs to find a vector \(\mathbf{v}\) such that \(\mathbf{v} imes egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\), where the cross product is defined in 3D. Find a suitable \(\mathbf{v}\).

📌 Let \(\mathbf{v} = egin{pmatrix} x \ y \ z \end{pmatrix}\), \(\mathbf{a} = egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}\), and \(\mathbf{v} imes \mathbf{a} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\).

📌 \mathbf{v} imes \mathbf{a} = egin{vmatrix}

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x^4 + 3x^3 - 2x^2 + x + 5 = (x^2 - x + 1)(x^2 + ax + b) + ax + c
Expand right-hand side:
= x^4 + (a - 1)x^3 + (b - a + 1)x^2 + (-b + a)x + b
Add \(ax + c\):
x^4 + (a - 1)x^3 + (b - a + 1)x^2 + (-b + a + a)x + (b + c) = x^4 + (a...
Match coefficients:
\(x^4\): 1 = 1 â
\(x^3\): \(a - 1 = 3 \Rightarrow a = 4\)
\(x^2\): \(b - a + 1 = -2\), \(b - 4 + 1 = -2 \Rightarrow b = 1\)
\(x\): \(2a - b = 1 \Rightarrow 8 - 1 = 7

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