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Home » This Jolly Nature Pokémon Will Make You Smile—Find It Before It Vanishes! - Groen Casting

This Jolly Nature Pokémon Will Make You Smile—Find It Before It Vanishes! - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 But since we already computed it:

📌 oxed{\dfrac{5\sqrt{2}}{2}}

📌 A materials scientist is studying the symmetry of molecular arrangements in a self-healing material. The arrangement has a rotational symmetry group isomorphic to the dihedral group \(D_4\). How many distinct vectors \(\mathbf{v} = egin{pmatrix} x \ y \end{pmatrix}\) in \(\mathbb{R}^2\) with integer components satisfy \( \mathbf{v} \cdot \mathbf{v} = 10 \) and are invariant under a \(90^\circ\) rotation about the origin?

📌 The standard \(90^\circ\) rotation matrix is:

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

📌 A vector \(\mathbf{v} = egin{pmatrix} x \ y \end{pmatrix}\) is invariant under \(R\) if:

📌 R\mathbf{v} = egin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} egin{pmatrix} x \ y \end{pmatrix} = egin{pmatrix} -y \ x \end{pmatrix} = egin{pmatrix} x \ y \end{pmatrix}

📌 -y = x \quad ext{and} \quad x = y

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