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 $\mathbf{b}$.

["Title: Modeling Stellar Trajectories Near Black Holes Using Orthogonal Vectors in Astrophysics", "In the realm of astrophysics, modeling the motion of stars near black holes is a critical challenge—especially when determining how gravitational forces influence a star’s trajectory. A powerful mathematical tool in such modeling is vector analysis, particularly finding a vector orthogonal to two given vectors. This concept becomes essential in simulating orbital planes and gravitational lensing effects near massive compact objects.", "Consider two vectors $\mathbf{a} = \begin{pmatrix} 2 \ -3 \ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \ 4 \ -2 \end{pmatrix}$ representing key directional components in a star’s velocity and gravitational influence fields. To find a vector $\mathbf{c}$ orthogonal to both $\mathbf{a}$ and $\mathbf{b}$, researchers use the cross product—a fundamental operation in vector calculus.", "### The Cross Product: Finding the Orthogonal Vector", "For vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^3$, their cross product is defined as:\n$$\n\mathbf{a} \ imes \mathbf{b} = \begin{vmatrix}\n\mathbf{i} & \mathbf{j} & \mathbf{k} \\n2 & -3 & 1 \\n1 & 4 & -2 \\n\end{vmatrix}\n$$", "Computing the determinant:\n$$\n\mathbf{a} \ imes \mathbf{b} = \mathbf{i} \left( (-3)(-2) - (1)(4) \right) - \mathbf{j} \left( (2)(-2) - (1)(1) \right) + \mathbf{k} \left( (2)(4) - (-3)(1) \right)\n$$\n$$\n= \mathbf{i}(6 - 4) - \mathbf{j}(-4 - 1) + \mathbf{k}(8 + 3)\n$$\n$$\n= 2\mathbf{i} + 5\mathbf{j} + 11\mathbf{k} = \begin{pmatrix} 2 \ 5 \ 11 \end{pmatrix}\n$$", "Thus, the vector $\mathbf{c} = \begin{pmatrix} 2 \ 5 \ 11 \end{pmatrix}$ is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$, satisfying the condition $\mathbf{a} \cdot \mathbf{c} = 0$ and $\mathbf{b} \cdot \mathbf{c} = 0$.", "### Application in Astrophysical Modeling", "In astrophysical simulations, such orthogonal vectors help define coordinate axes that are aligned with the star’s local orbital plane. When modeling trajectories near black holes, where spacetime curvature dominates, using precisely oriented vectors ensures accurate vector-based predictions of motion, light deflection, and energy transfer. The cross product provides a computationally efficient and geometrically meaningful method to generate these reference vectors.", "Conclusion\nBy leveraging the cross product, astrophysics researchers can robustly model the influence of gravitational fields through vector orthogonality. The vector $\begin{pmatrix} 2 \ 5 \ 11 \end{pmatrix}$ serves as a key mathematical construct in simulating how a star’s path curves near a black hole—enhancing both precision and clarity in theoretical and computational astrophysics.", "---\nStay tuned for more insights into how vector mathematics drives cutting-edge discoveries in black hole dynamics and stellar mechanics."]