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how long do kidney stones last - GetMeFoodie

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 \boxed{30^\circ, 90^\circ, 150^\circ}

📌 Question: Find the value of $x$ so that the vectors $\begin{pmatrix} 2 \\ x \\ 3 \end{pmatrix}$ and $\begin{pmatrix} x \\ -1 \\ 4 \end{pmatrix}$ are orthogonal.

📌 Two vectors are orthogonal if their dot product is zero:

📌 \begin{pmatrix} 2 \\ x \\ 3 \end{pmatrix} \cdot \begin{pmatrix} x \\ -1 \\ 4 \end{pmatrix} = 2x + x(-1) + 3(4) = 2x - x + 12 = x + 12 = 0

📌 x + 12 = 0 \Rightarrow x = \boxed{-12}

📌 Question: Find the matrix $\mathbf{M}$ such that

📌 \mathbf{M} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \quad \mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}

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