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hilton beachfront resort and spa hilton head island - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 \boxed{\frac{\sqrt{5}}{5}}

📌 An astrophysics researcher models the trajectory of a star near a black hole using vectors. If $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}$, find a vector $\mathbf{c}$ orthogonal to both $\mathbf{a}$ and $\mathbf{b}$.

📌 A vector $\mathbf{c}$ orthogonal to both $\mathbf{a}$ and $\mathbf{b}$ can be found using the cross product:

📌 \mathbf{c} = \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -3 & 1 \\ 1 & 4 & -2 \end{vmatrix}

📌 Calculate the determinant:

📌 \mathbf{c} = \mathbf{i}((-3)(-2) - (1)(4)) - \mathbf{j}((2)(-2) - (1)(1)) + \mathbf{k}((2)(4) - (-3)(1))

📌 = \mathbf{i}(6 - 4) - \mathbf{j}(-4 - 1) + \mathbf{k}(8 + 3)

📌 = \mathbf{i}(2) + \mathbf{j}(5) + \mathbf{k}(11)

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