Let me write the first question:

February 22, 2026

Let me write the first question:

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Fourth: Range of a function representing precipitation patterns, with domain restrictions.
Fifth: Vector OC as a combination of OA and OB, contextualized in climate model vectors.
Now, structure each with Question: and Solution: sections, ensuring they’re distinct and fit the personas. Check for mathematical accuracy and relevance to the fields. Make sure solutions are detailed and correct. Avoid any markdown, just plain text with LaTeX.
Question: Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{w} = \mathbf{p}$, where $\mathbf{w} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ represents wind velocity and $\mathbf{p} = \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix}$ models the pollination force vector.
Solution: To solve for $\mathbf{v}$, we use the cross product equation $\mathbf{v} \times \mathbf{w} = \mathbf{p}$. Let $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$. The cross product $\mathbf{v} \times \mathbf{w}$ is calculated as $\begin{pmatrix} v_2 \cdot 3 - v_3 \cdot (-1) \\ v_3 \cdot 2 - v_1 \cdot 3 \\ v_1 \cdot (-1) - v_2 \cdot 2 \end{pmatrix} = \begin{pmatrix} 3v_2 + v_3 \\ 2v_3 - 3v_1 \\ -v_1 - 2v_2 \end{pmatrix}$. Setting this equal to $\mathbf{p} = \begin{pmatrix} 5
3v_2 + v_3 = 5 \\

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