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Home » From Secret Origins to Stardom: Alena Croft’s Rise You Won’t Stop Talking About! - Crankk.io

From Secret Origins to Stardom: Alena Croft’s Rise You Won’t Stop Talking About! - Crankk.io

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Solve for $\cos \theta$:

📌 \cos \theta = \frac{5}{5 \cdot \sqrt{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}

📌 \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))

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